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{{about|the mathematical constant|the Greek letter|Pi (letter)|other uses}}{{pp-protect|small=yes}}{{short description|Ratio of the circumference of a circle to its diameter}}{{Use dmy dates|date=July 2012}}{{Pi box}}The number {{pi}} ({{IPAc-en|p|aÉª}}) is a mathematical constant. Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "{{pi}}" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes' constant.Being an irrational number, {{pi}} cannot be expressed as a common fraction (equivalently, its decimal representation never ends and never settles into a permanently repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate {{pi}}. The digits appear to be randomly distributed. In particular, the digit sequence of {{pi}} is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. Also, {{pi}} is a transcendental number; that is, it is not the root of any polynomial having rational coefficients. This transcendence of {{pi}} implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.Ancient civilizations required fairly accurate computed values to approximate {{pi}} for practical reasons, including the Egyptians and Babylonians. Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. In the 5th century AD Chinese mathematics approximated {{pi}} to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The historically first exact formula for {{pi}}, based on infinite series, was not available until a millennium later, when in the 14th century the Madhavaâ€“Leibniz series was discovered in Indian mathematics.BOOK, Special Functions, George E., Andrews, Richard, Askey, Ranjan, Roy, Cambridge University Press, 1999, 978-0-521-78988-2, 58, JOURNAL, R.C., Gupta, On the remainder term in the Madhavaâ€“Leibniz's series, Ganita Bharati, 14, 1â€“4, 1992, 68â€“71, In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of {{pi}} to many trillions of digits after the decimal point.Ï€e trillion digits of Ï€ {{webarchive|url=https://web.archive.org/web/20161206063441weblink |date=6 December 2016 }} Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedesâ€™ constant on Google Cloud Practically all scientific applications require no more than a few hundred digits of {{pi}}, and many substantially fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the desire to break records.{{harvnb|Arndt|Haenel|2006|p=17}}JOURNAL, David, Bailey, Jonathan, Borwein, Peter, Borwein, Simon, Plouffe, The Quest for Pi, The Mathematical Intelligencer, 1997, 19, 1, 50â€“56, 10.1007/bf03024340, 10.1.1.138.7085, The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.Because its most elementary definition relates to the circle, {{pi}} is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics, as well as in almost all areas of physics. The ubiquity of {{pi}} makes it one of the most widely known mathematical constants both inside and outside the scientific community. Several books devoted to {{pi}} have been published, and record-setting calculations of the digits of {{pi}} often result in news headlines. Attempts to memorize the value of {{pi}} with increasing precision have led to records of over 70,000 digits.{{TOC limit|limit=3}}

Fundamentals

Name

The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter {{pi}}, sometimes spelled out as pi, and derived from the first letter of the Greek word perimetros, meaning circumference.JOURNAL, Boeing, Niels, 14 March 2016,weblink Die Welt ist Pi, The World is Pi, de, Zeit Online, Die Ludolphsche Zahl oder Kreiszahl erhielt nun auch das Symbol, unter dem wir es heute kennen: William Jones schlug 1706 den griechischen Buchstaben Ï€ vor, in Anlehnung an perimetros, griechisch fÃ¼r Umfang. Leonhard Euler etablierte Ï€ schlieÃŸlich in seinen mathematischen Schriften. [The Ludolphian number or circle number now also received the symbol under which we know it today: William Jones proposed in 1706 the Greek letter Ï€, based on perimetros [Ï€ÎµÏÎ¯Î¼ÎµÏ„ÏÎ¿Ï‚], Greek for perimeter. Leonhard Euler firmly established Ï€ in his mathematical writings.], live,weblink" title="web.archive.org/web/20160317050313weblink">weblink 17 March 2016, dmy-all, In English, {{pi}} is pronounced as "pie" ({{IPAc-en|p|aÉª}} {{respell|PY}}).WEB,weblink pi, Dictionary.reference.com, 2 March 1993, 18 June 2012, live,weblink" title="web.archive.org/web/20140728121603weblink">weblink 28 July 2014, dmy-all, In mathematical use, the lowercase letter {{pi}} (or Ï€ in sans-serif font) is distinguished from its capitalized and enlarged counterpart {{math|âˆ}}, which denotes a product of a sequence, analogous to how {{math|âˆ‘}} denotes summation.The choice of the symbol {{pi}} is discussed in the section Adoption of the symbol {{pi}}.

Definition

(File:Pi eq C over d.svg|alt=A diagram of a circle, with the width labeled as diameter, and the perimeter labeled as circumference|thumb|right|The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called {{pi}}.){{pi}} is commonly defined as the ratio of a circle's circumference {{math|C}} to its diameter {{math|d}}:{{harvnb|Arndt|Haenel|2006|p=8}}
pi = frac{C}{d}.
The ratio {{math|C/d}} is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio {{math|C/d}}. This definition of {{pi}} implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (non-Euclidean) geometry, these new circles will no longer satisfy the formula {{math|{{pi}} {{=}} C/d}}.Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus.BOOK, Tom, Apostol, Tom Apostol, Calculus, volume 1, Wiley, 2nd, 1967, . p. 102: "From a logical point of view, this is unsatisfactory at the present stage because we have not yet discussed the concept of arc length." Arc length is introduced on p. 529. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation {{math|x2 + y2 {{=}} 1}}, as the integral:{{citation|first=Reinhold|last=Remmert|chapter=What is {{pi}}?|title=Numbers|publisher=Springer|year=1991|page=129}}
pi = int_{-1}^1 frac{dx}{sqrt{1-x^2}}.
An integral such as this was adopted as the definition of {{pi}} by Karl Weierstrass, who defined it directly as an integral in 1841.{{harvtxt|Remmert|1991}}. The precise integral that Weierstrass used was pi=int_{-infty}^inftyfrac{dx}{1+x^2}.Definitions of {{pi}} such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. {{harvtxt|Remmert|1991}} explains that this is because in many modern treatments of calculus, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of {{pi}} that does not rely on the latter. One such definition, due to Richard Baltzer,{{citation|first=Richard |last=Baltzer |authorlink=Richard Baltzer |title=Die Elemente der Mathematik |language=de |trans-title=The Elements of Mathematics |year=1870 |page=195 |url=https://archive.org/details/dieelementederm02baltgoog |publisher=Hirzel |url-status=live |archiveurl=https://web.archive.org/web/20160914204826weblink |archivedate=14 September 2016 |df=dmy-all}} and popularized by Edmund Landau,{{citation|first=Edmund |last=Landau |authorlink=Edmund Landau |title=EinfÃ¼hrung in die Differentialrechnung und Integralrechnung |language=de |publisher=Noordoff |year=1934 |page=193}} is the following: {{pi}} is twice the smallest positive number at which the cosine function equals 0.BOOK, Rudin, Walter, Principles of Mathematical Analysis, McGraw-Hill, 1976, 978-0-07-054235-8, harv, , p. 183. The cosine can be defined independently of geometry as a power series,BOOK, Rudin, Walter, Real and complex analysis, McGraw-Hill, 1986, harv, , p. 2. or as the solution of a differential equation.In a similar spirit, {{pi}} can be defined instead using properties of the complex exponential, {{math|exp z}}, of a complex variable {{math|z}}. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which {{math|exp z}} is equal to one is then an (imaginary) arithmetic progression of the form:
{dots,-2pi i, 0, 2pi i, 4pi i,dots} = {2pi kimid kinmathbb Z}
and there is a unique positive real number {{pi}} with this property.{{citation|first=Lars |last=Ahlfors |authorlink=Lars Ahlfors |title=Complex analysis |publisher=McGraw-Hill |year=1966 |p=46}}A more abstract variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem:{{citation|last=Bourbaki |first=Nicolas |authorlink=Nicolas Bourbaki |title=Topologie generale |publisher=Springer |year=1981}}, Â§VIII.2. there is a unique (up to automorphism) continuous isomorphism from the group R/Z of real numbers under addition modulo integers (the circle group) onto the multiplicative group of complex numbers of absolute value one. The number {{pi}} is then defined as half the magnitude of the derivative of this homomorphism.{{citation |last=Bourbaki |first=Nicolas |authorlink=Nicolas Bourbaki |title=Fonctions d'une variable rÃ©elle |language=fr |publisher=Springer |year=1979}}, Â§II.3.A circle encloses the largest area that can be attained within a given perimeter. Thus the number {{pi}} is also characterized as the best constant in the isoperimetric inequality (times one-fourth). There are many other, closely related, ways in which {{pi}} appears as an eigenvalue of some geometrical or physical process; see below.

Irrationality and normality

{{pi}} is an irrational number, meaning that it cannot be written as the ratio of two integers (fractions such as {{math|{{sfrac|22|7}}}} and {{math|{{sfrac|355|113}}}} are commonly used to approximate {{pi}}, but no common fraction (ratio of whole numbers) can be its exact value).{{harvnb|Arndt|Haenel|2006|p=5}} Because {{pi}} is irrational, it has an infinite number of digits in its decimal representation, and it does not settle into an infinitely repeating pattern of digits. There are several proofs that {{pi}} is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which {{pi}} can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of {{math|e}} or {{math|ln 2}} but smaller than the measure of Liouville numbers.JOURNAL, Salikhov, V., 2008, On the Irrationality Measure of pi, Russian Mathematical Surveys, 53, 3, 570â€“572, harv, 10.1070/RM2008v063n03ABEH004543, 2008RuMaS..63..570S, The digits of {{pi}} have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that {{pi}} is normal has not been proven or disproven.{{harvnb|Arndt|Haenel|2006|pp=22â€“23}}NEWS,weblink Are The Digits of Pi Random? Lab Researcher May Hold The Key, Paul, Preuss, Paul Preuss (author), Lawrence Berkeley National Laboratory, 23 July 2001, 10 November 2007, live,weblink" title="web.archive.org/web/20071020010208weblink">weblink 20 October 2007, dmy-all, Since the advent of computers, a large number of digits of {{pi}} have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of {{pi}} and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.{{harvnb|Arndt|Haenel|2006|pp=22, 28â€“30}} Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem. Thus, because the sequence of {{pi}}'s digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of {{pi}}.{{harvnb|Arndt|Haenel|2006|p=3}} This is also called the "Feynman point" in mathematical folklore, after Richard Feynman, although no connection to Feynman is known.

Transcendence

File:Squaring the circle.svg|thumb|alt=A diagram of a square and circle, both with identical area; the length of the side of the square is the square root of pi|Because {{pi}} is a transcendental number, squaring the circle is not possible in a finite number of steps using the classical tools of compass and straightedge.]]In addition to being irrational, more strongly {{pi}} is a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as {{math|{{sfrac|x5|120}} âˆ’ {{sfrac|x3|6}} + x {{=}} 0}}.WEB, Steve, Mayer,weblinkweblink" title="web.archive.org/web/20000929033317weblink">weblink 2000-09-29, The Transcendence of {{pi, |accessdate=4 November 2007}}The polynomial shown is the first few terms of the Taylor series expansion of the sine function.The transcendence of {{pi}} has two important consequences: First, {{pi}} cannot be expressed using any finite combination of rational numbers and square roots or n-th roots such as {{math|{{radic|31|3}}}} or {{math|{{sqrt|10}}}}. Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle.{{harvnb|Posamentier|Lehmann|2004|p=25}} Squaring a circle was one of the important geometry problems of the classical antiquity.{{harvnb|Eymard|Lafon|1999|p=129}} Amateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is mathematically impossible.{{harvnb|Beckmann|1989|p=37}}BOOK, Schlager, Neil, Lauer, Josh, Science and Its Times: Understanding the Social Significance of Scientific Discovery, Gale Group, 2001, 978-0-7876-3933-4, harv, , p. 185.

Continued fractions

File:Matheon2.jpg|thumb|alt=A photograph of the Greek letter pi, created as a large stone mosaic embedded in the ground.|The constant {{pi}} is represented in this mosaic outside the Mathematics Building at the Technical University of BerlinTechnical University of BerlinLike all irrational numbers, {{pi}} cannot be represented as a common fraction (also known as a simple or vulgar fraction), by the very definition of "irrational number" (that is, "not a rational number"). But every irrational number, including {{pi}}, can be represented by an infinite series of nested fractions, called a continued fraction:
pi=3+textstyle cfrac{1}{7+textstyle cfrac{1}{15+textstyle cfrac{1}{1+textstyle cfrac{1}{292+textstyle cfrac{1}{1+textstyle cfrac{1}{1+textstyle cfrac{1}{1+ddots}}}}}}}Truncating the continued fraction at any point yields a rational approximation for {{pi}}; the first four of these are 3, 22/7, 333/106, and 355/113. These numbers are among the most well-known and widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to {{pi}} than any other fraction with the same or a smaller denominator.{{harvnb|Eymard|Lafon|1999|p=78}} Because {{pi}} is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, {{pi}} cannot have a periodic continued fraction. Although the simple continued fraction for {{pi}} (shown above) also does not exhibit any other obvious pattern,OEIS, A001203, Continued fraction for Pi, Retrieved 12 April 2012. mathematicians have discovered several generalized continued fractions that do, such as:JOURNAL, An Elegant Continued Fraction for {{pi, |first=L.J.|last=Lange|journal=The American Mathematical Monthly|volume=106|issue=5| date=May 1999 |pages=456â€“458|jstor=2589152|doi=10.2307/2589152|ref=harv}}
begin{align}pi & = textstyle cfrac{4}{1+textstyle cfrac{1^2}{2+textstyle cfrac{3^2}{2+textstyle cfrac{5^2}{2+textstyle cfrac{7^2}{2+textstyle cfrac{9^2}{2+ddots}}}}}}

3+textstyle cfrac{1^2}{6+textstyle cfrac{3^2}{6+textstyle cfrac{5^2}{6+textstyle cfrac{7^2}{6+textstyle cfrac{9^2}{6+ddots}}}}} [8pt]

& =textstyle cfrac{4}{1+textstyle cfrac{1^2}{3+textstyle cfrac{2^2}{5+textstyle cfrac{3^2}{7+textstyle cfrac{4^2}{9+ddots}}}}}end{align}

Approximate value and digits

Some approximations of pi include:
• Integers: 3
• Fractions: Approximate fractions include (in order of increasing accuracy) {{sfrac|22|7}}, {{sfrac|333|106}}, {{sfrac|355|113}}, {{sfrac|52163|16604}}, {{sfrac|103993|33102}}, {{sfrac|104348|33215}}, and {{sfrac|245850922|78256779}}. (List is selected terms from {{OEIS2C|id=A063674}} and {{OEIS2C|id=A063673}}.)
• Digits: The first 50 decimal digits are {{gaps|3.14159|26535|89793|23846|26433|83279|50288|41971|69399|37510...}}{{harvnb|Arndt|Haenel|2006|p=240}} (see {{OEIS2C|id=A000796}})
Digits in other number systems
• The first 48 binary (base 2) digits (called bits) are {{gaps|11.0010|0100|0011|1111|0110|1010|1000|1000|1000|0101|1010|0011...}} (see {{OEIS2C|id=A004601}})
• The first 20 digits in hexadecimal (base 16) are {{gaps|3.243F|6A88|85A3|08D3|1319...}}{{harvnb|Arndt|Haenel|2006|p=242}} (see {{OEIS2C|id=A062964}})
• The first five sexagesimal (base 60) digits are 3;8,29,44,0,47{{citation|title=Abu-r-Raihan al-Biruni, 973â€“1048|last=Kennedy|first=E.S.|journal=Journal for the History of Astronomy|volume=9|page=65|bibcode=1978JHA.....9...65K|doi=10.1177/002182867800900106|year=1978}}. Ptolemy used a three-sexagesimal-digit approximation, and JamshÄ«d al-KÄshÄ« expanded this to nine digits; see {{Citation |last= Aaboe |first= Asger |authorlink= Asger Aaboe |year= 1964 |title= Episodes from the Early History of Mathematics |series= New Mathematical Library |volume= 13 |publisher= Random House |location= New York |page= 125 |url=weblink |url-status=live |archiveurl=weblink |archivedate= 29 November 2016 |df= dmy-all |isbn= 978-0-88385-613-0 }} (see {{OEIS2C|id=A060707}})

Complex numbers and Euler's identity

File:Euler's formula.svg|thumb|alt=A diagram of a unit circle centered at the origin in the complex plane, including a ray from the center of the circle to its edge, with the triangle legs labeled with sine and cosine functions.|The association between imaginary powers of the number {{math|e}} and points on the unit circle centered at the origin in the complex plane given by Euler's formulaEuler's formulaAny complex number, say {{math|z}}, can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or r) is used to represent {{math|z}}'s distance from the origin of the complex plane and the other (angle or {{math|Ï†}}) to represent a counter-clockwise rotation from the positive real line as follows:{{harvnb|Ayers|1964|p=100}}
z = rcdot(cosvarphi + isinvarphi),
where {{math|i}} is the imaginary unit satisfying {{math|i2}} = âˆ’1. The frequent appearance of {{pi}} in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula:{{harvnb|BronshteÄ­n|Semendiaev|1971|p=592}}
e^{ivarphi} = cos varphi + isin varphi,
where the constant {{math|e}} is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of {{math|e}} and points on the unit circle centered at the origin of the complex plane. Setting {{math|Ï†}} = {{pi}} in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants:Maor, Eli, E: The Story of a Number, Princeton University Press, 2009, p. 160, {{isbn|978-0-691-14134-3}} ("five most important" constants).
e^{i pi} + 1 = 0.
There are {{math|n}} different complex numbers {{math|z}} satisfying {{math|1=zn = 1}}, and these are called the "{{math|n}}-th roots of unity".{{MathWorld|RootofUnity|Roots of Unity}} They are given by this formula:
e^{2 pi i k/n} qquad (k = 0, 1, 2, dots, n - 1).

History

Approximations of {{pi}}}}{{see also|Chronology of computation of Ï€|l1=Chronology of computation of {{pi}}}}

Infinite series

{{comparison_pi_infinite_series.svg}}The calculation of {{pi}} was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.{{harvnb|Arndt|Haenel|2006|pp=185â€“191}} Infinite series allowed mathematicians to compute {{pi}} with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for {{pi}} most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach was first discovered in India sometime between 1400 and 1500 AD.{{harvnb|Roy|1990|pp=101â€“102}}{{harvnb|Arndt|Haenel|2006|pp=185â€“186}} The first written description of an infinite series that could be used to compute {{pi}} was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.{{harvnb|Roy|1990|pp=101â€“102}} The series are presented without proof, but proofs are presented in a later Indian work, YuktibhÄá¹£Ä, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 â€“ c. 1425. Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregoryâ€“Leibniz series. Madhava used infinite series to estimate {{pi}} to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician JamshÄ«d al-KÄshÄ«, using a polygonal algorithm.{{harvnb|Joseph|1991|p=264}}File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|upright|alt=A formal portrait of a man, with long hair|Isaac Newtonused infinite seriesinfinite seriesThe first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in {{pi}} calculations) found by French mathematician FranÃ§ois ViÃ¨te in 1593:{{harvnb|Arndt|Haenel|2006|p=187}}{{OEIS2C|id=A060294}}Variorum de rebus mathematicis responsorum liber VIII.
frac2pi = frac{sqrt2}2 cdot frac{sqrt{2+sqrt2}}2 cdot frac{sqrt{2+sqrt{2+sqrt2}}}2 cdots
The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product:
frac{pi}{2} = Big(frac{2}{1} cdot frac{2}{3}Big) cdot Big(frac{4}{3} cdot frac{4}{5}Big) cdot Big(frac{6}{5} cdot frac{6}{7}Big) cdot Big(frac{8}{7} cdot frac{8}{9}Big) cdotsThe discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating {{pi}}. Newton himself used an arcsin series to compute a 15 digit approximation of {{pi}} in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."{{harvnb|Arndt|Haenel|2006|p=188}}. Newton quoted by Arndt.In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregory in 1671, and by Leibniz in 1674:{{harvnb|Arndt|Haenel|2006|pp=188â€“189}}
arctan z = z - frac {z^3} {3} +frac {z^5} {5} -frac {z^7} {7} +cdotsThis formula, the Gregoryâ€“Leibniz series, equals {{math|Ï€/4}} when evaluated with {{math|z}} = 1.{{harvnb|Eymard|Lafon|1999|pp=53â€“54}} In 1699, English mathematician Abraham Sharp used the Gregoryâ€“Leibniz series for z=frac{1}{sqrt{3}} to compute {{pi}} to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.{{harvnb|Arndt|Haenel|2006|p=189}} The Gregoryâ€“Leibniz for z=1 series is simple, but converges very slowly (that is, approaches the answer gradually), so it is not used in modern {{pi}} calculations.{{harvnb|Arndt|Haenel|2006|p=156}}In 1706 John Machin used the Gregoryâ€“Leibniz series to produce an algorithm that converged much faster:{{harvnb|Arndt|Haenel|2006|pp=192â€“193}}
frac{pi}{4} = 4 arctan frac{1}{5} - arctan frac{1}{239}.
Machin reached 100 digits of {{pi}} with this formula.{{harvnb|Arndt|Haenel|2006|pp=72â€“74}} Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of {{pi}}. Machin-like formulae remained the best-known method for calculating {{pi}} well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson â€“ the best approximation achieved without the aid of a calculating device.{{harvnb|Arndt|Haenel|2006|pp=192â€“196, 205}}A record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of German mathematician Carl Friedrich Gauss. British mathematician William Shanks famously took 15 years to calculate {{pi}} to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.{{harvnb|Arndt|Haenel|2006|pp=194â€“196}}

Rate of convergence

Some infinite series for {{pi}} converge faster than others. Given the choice of two infinite series for {{pi}}, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate {{pi}} to any given accuracy.JOURNAL, Borwein, J.M., Borwein, P.B., Ramanujan and Pi, 1988, Scientific American, 256, 2, 112â€“117, harv, 1988SciAm.258b.112B, 10.1038/scientificamerican0288-112, {{harvnb|Arndt|Haenel|2006|pp=15â€“17, 70â€“72, 104, 156, 192â€“197, 201â€“202}} A simple infinite series for {{pi}} is the Gregoryâ€“Leibniz series:{{harvnb|Arndt|Haenel|2006|pp=69â€“72}}
pi = frac{4}{1} - frac{4}{3} + frac{4}{5} - frac{4}{7} + frac{4}{9} - frac{4}{11} + frac{4}{13} - cdots
As individual terms of this infinite series are added to the sum, the total gradually gets closer to {{pi}}, and â€“ with a sufficient number of terms â€“ can get as close to {{pi}} as desired. It converges quite slowly, though â€“ after 500,000 terms, it produces only five correct decimal digits of {{pi}}.JOURNAL, Borwein, J.M., Borwein, P.B., Dilcher, K., 1989, Pi, Euler Numbers, and Asymptotic Expansions, American Mathematical Monthly, 96, 8, 681â€“687, 10.2307/2324715, harv, 2324715, An infinite series for {{pi}} (published by Nilakantha in the 15th century) that converges more rapidly than the Gregoryâ€“Leibniz series is:{{harvnb|Arndt|Haenel|2006|p=223}}, (formula 16.10). Note that (n âˆ’ 1)n(n + 1) = n3 âˆ’ n.BOOK, Wells, David, 35, The Penguin Dictionary of Curious and Interesting Numbers, revised, Penguin, 1997, 978-0-14-026149-3, harv,
pi = 3 + frac{4}{2times3times4} - frac{4}{4times5times6} + frac{4}{6times7times8} - frac{4}{8times9times10} + cdots
The following table compares the convergence rates of these two series:{|class="wikitable" style="text-align: center; margin: auto;"! Infinite series for {{pi}} !! After 1st term !! After 2nd term !! After 3rd term !! After 4th term !! After 5th term !! Converges to:
| pi = frac{4}{1} - frac{4}{3} + frac{4}{5} - frac{4}{7} + frac{4}{9} - frac{4}{11} + frac{4}{13} + cdots
4.00002.6666 ... 3.4666 ... 2.8952 ... 3.3396 ... rowspan=2| {{pi}} = 3.1415 ...
| pi = {{3}} + frac{{4}}{2times3times4} - frac{{4}}{4times5times6} + frac{{4}}{6times7times8} + cdots
3.00003.1666 ... 3.1333 ... 3.1452 ... 3.1396 ...
After five terms, the sum of the Gregoryâ€“Leibniz series is within 0.2 of the correct value of {{pi}}, whereas the sum of Nilakantha's series is within 0.002 of the correct value of {{pi}}. Nilakantha's series converges faster and is more useful for computing digits of {{pi}}. Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term.

Irrationality and transcendence

{{See also|Proof that Ï€ is irrational{{!}}Proof that {{pi}} is irrational|Proof that Ï€ is transcendental{{!}}Proof that {{pi}} is transcendental}}Not all mathematical advances relating to {{pi}} were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between {{pi}} and the prime numbers that later contributed to the development and study of the Riemann zeta function:{{harvnb|Posamentier|Lehmann|2004|pp=284}}
frac{pi^2}{6} = frac{1}{1^2} + frac{1}{2^2} + frac{1}{3^2} + frac{1}{4^2} + cdots
Swiss scientist Johann Heinrich Lambert in 1761 proved that {{pi}} is irrational, meaning it is not equal to the quotient of any two whole numbers. Lambert's proof exploited a continued-fraction representation of the tangent function.Lambert, Johann, "MÃ©moire sur quelques propriÃ©tÃ©s remarquables des quantitÃ©s transcendantes circulaires et logarithmiques", reprinted in {{harvnb|Berggren|Borwein|Borwein|1997|pp=129â€“140}} French mathematician Adrien-Marie Legendre proved in 1794 that {{pi}}2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that {{pi}} is transcendental, confirming a conjecture made by both Legendre and Euler.{{harvnb|Arndt|Haenel|2006|p=196}}.Hardy and Wright 1938 and 2000: 177 footnote Â§11.13â€“14 references Lindemann's proof as appearing at Math. Ann. 20 (1882), 213â€“225. Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".cf Hardy and Wright 1938 and 2000:177 footnote Â§11.13â€“14. The proofs that e and Ï€ are transcendental can be found on pp. 170â€“176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pp. 417â€“419 for full citations.

Modern quest for more digits

Computer era and iterative algorithms

File:JohnvonNeumann-LosAlamos.gif|thumb|upright|alt=Formal photo of a balding man wearing a suit|John von Neumann was part of the team that first used a digital computer, ENIACENIAC
is given by

scriptstyle pi approx frac{(a_n + b_n)^2}{4 t_n}.qalign=left}}

Motives for computing {{pi}}

File:Record pi approximations.svg|thumb|400px|right|As mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of {{pi}} increased dramatically. Note that the vertical scale is logarithmlogarithmFor most numerical calculations involving {{pi}}, a handful of digits provide sufficient precision. According to JÃ¶rg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the circumference of the observable universe with a precision of one atom.{{citation|title=Pi and the size of the Universe|author=James Grime|publisher=Numberphile|url=https://www.youtube.com/watch?v=FpyrF_Ci2TQ}} Accounting for additional digits needed to compensate for computational round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to compute {{pi}} to thousands and millions of digits.{{harvnb|Arndt|Haenel|2006|pp=17â€“19}} This effort may be partly ascribed to the human compulsion to break records, and such achievements with {{pi}} often make headlines around the world.NEWS, John W. Wrench, Jr.: Mathematician Had a Taste for Pi, Matt, Schudel, The Washington Post, 25 March 2009, B5, NEWS, The Big Question: How close have we come to knowing the precise value of pi?,weblink The Independent, 8 January 2010, 14 April 2012, London, Steve, Connor, live,weblink" title="web.archive.org/web/20120402220803weblink">weblink 2 April 2012, dmy-all, They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of {{pi}}.{{harvnb|Arndt|Haenel|2006|p=18}}

Rapidly convergent series

File:Srinivasa Ramanujan - OPC - 1.jpg|thumb|upright|alt=Photo portrait of a man| Srinivasa RamanujanSrinivasa RamanujanModern {{pi}} calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive. The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for {{pi}}, remarkable for their elegance, mathematical depth, and rapid convergence.{{harvnb|Arndt|Haenel|2006|pp=103â€“104}} One of his formulae, based on modular equations, is
frac{1}{pi} = frac{2 sqrt 2}{9801} sum_{k=0}^infty frac{(4k)!(1103+26390k)}{k!^4left(396^{4k}right)}.
This series converges much more rapidly than most arctan series, including Machin's formula.{{harvnb|Arndt|Haenel|2006|p=104}} Bill Gosper was the first to use it for advances in the calculation of {{pi}}, setting a record of 17 million digits in 1985.{{harvnb|Arndt|Haenel|2006|pp=104, 206}} Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.{{harvnb|Arndt|Haenel|2006|pp=110â€“111}} The Chudnovsky formula developed in 1987 is
frac{1}{pi} = frac{12}{640320^{3/2}} sum_{k=0}^infty frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}.
It produces about 14 digits of {{pi}} per term,{{harvnb|Eymard|Lafon|1999|p=254}} and has been used for several record-setting {{pi}} calculations, including the first to surpass 1 billion (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7Ã—1012) digits by Fabrice Bellard in 2009,{{harvnb|Arndt|Haenel|2006|pp=110â€“111, 206}}Bellard, Fabrice, "Computation of 2700 billion decimal digits of Pi using a Desktop Computer" {{webarchive|url=https://web.archive.org/web/20110518160217weblink |date=18 May 2011 }}, 11 Feb 2010. 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo,"Round 2... 10 Trillion Digits of Pi" {{webarchive|url=https://web.archive.org/web/20140101121120weblink |date=1 January 2014 }}, NumberWorld.org, 17 Oct 2011. Retrieved 30 May 2012. and over 22 trillion digits in 2016 by Peter Trueb.NEWS, Timothy Revell, Celebrate pi day with 9 trillion more digits than ever before,weblink New Scientist, 14 March 2017, 6 September 2018, HTTP://WWW.NUMBERWORLD.ORG/DIGITS/PI/ > TITLE=PI, 6 September 2018, For similar formulas, see also the Ramanujanâ€“Sato series.In 2006, mathematician Simon Plouffe used the PSLQ integer relation algorithmPSLQ means Partial Sum of Least Squares. to generate several new formulas for {{pi}}, conforming to the following template:
pi^k = sum_{n=1}^infty frac{1}{n^k} left(frac{a}{q^n-1} + frac{b}{q^{2n}-1} + frac{c}{q^{4n}-1}right),
where {{math|q}} is {{math|eπ}} (Gelfond's constant), {{math|k}} is an odd number, and {{math|a, b, c}} are certain rational numbers that Plouffe computed.WEB, Simon, Plouffe, Simon Plouffe, Identities inspired by Ramanujan's Notebooks (part 2), April 2006,weblink 10 April 2009, live,weblink" title="web.archive.org/web/20120114101641weblink">weblink 14 January 2012, dmy-all,

Monte Carlo methods

{{multiple image|direction=horizontal|footer=Monte Carlo methods, based on random trials, can be used to approximate {{pi}}.|width1=166|image1=Buffon needle.svg|caption1=Buffon's needle. Needles a and b are dropped randomly.|alt1=Needles of length â„“ scattered on stripes with width t|width2=100|caption2=Random dots are placed on the quadrant of a square with a circle inscribed in it.|alt2=Thousands of dots randomly covering a square and a circle inscribed in the square.}}Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of {{pi}}.{{harvnb|Arndt|Haenel|2006|p=39}} Buffon's needle is one such technique: If a needle of length {{math|â„“}} is dropped {{math|n}} times on a surface on which parallel lines are drawn {{math|t}} units apart, and if {{math|x}} of those times it comes to rest crossing a line ({{math|x}} > 0), then one may approximate {{pi}} based on the counts:JOURNAL, Ramaley, J.F., Buffon's Noodle Problem, 2317945, The American Mathematical Monthly, 76, 8, October 1969, 916â€“918, 10.2307/2317945, harv,
pi approx frac{2nell}{xt}.
Another Monte Carlo method for computing {{pi}} is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal {{math|Ï€/4}}.{{harvnb|Arndt|Haenel|2006|pp=39â€“40}}{{harvnb|Posamentier|Lehmann|2004|p=105}}(File:Five random walks.png|thumb|right|Five random walks with 200 steps. The sample mean of {{math|{{abs|W200}}}} is {{math|Î¼ {{=}} 56/5}}, and so {{math|2(200)Î¼âˆ’2 â‰ˆ 3.19}} is within {{math|0.05}} of {{pi}}.)Another way to calculate {{pi}} using probability is to start with a random walk, generated by a sequence of (fair) coin tosses: independent random variables {{math|Xk}} such that {{math|Xk âˆˆ {{mset|âˆ’1,1}}}} with equal probabilities. The associated random walk is
W_n = sum_{k=1}^n X_k
so that, for each {{mvar|n}}, {{math|Wn}} is drawn from a shifted and scaled binomial distribution. As {{mvar|n}} varies, {{math|Wn}} defines a (discrete) stochastic process. Then {{pi}} can be calculated by{{citation|last=GrÃ¼nbaum |first=B. |authorlink=Branko GrÃ¼nbaum|title=Projection Constants|journal=Trans. Amer. Math. Soc.|volume=95|issue=3 |pages=451â€“465|year=1960|doi=10.1090/s0002-9947-1960-0114110-9}}
pi = lim_{ntoinfty} frac{2n}{E[|W_n|]^2}.
This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below.These Monte Carlo methods for approximating {{pi}} are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they are never used to approximate {{pi}} when speed or accuracy is desired.{{harvnb|Arndt|Haenel|2006|pp=43}}{{harvnb|Posamentier|Lehmann|2004|pp=105â€“108}}

Spigot algorithms

Two algorithms were discovered in 1995 that opened up new avenues of research into {{pi}}. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of {{pi}} that are not reused after they are calculated.Gibbons, Jeremy, "Unbounded Spigot Algorithms for the Digits of Pi" {{webarchive|url=https://web.archive.org/web/20131202202928weblink |date=2 December 2013 }}, 2005. Gibbons produced an improved version of Wagon's algorithm. This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.{{harvnb|Arndt|Haenel|2006|pp=77â€“84}}Mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995.{{harvnb|Arndt|Haenel|2006|p=77}}JOURNAL, Stanley, Rabinowitz, Wagon, Stan, March 1995, A spigot algorithm for the digits of Pi, American Mathematical Monthly, 102, 3, 195â€“203, 10.2307/2975006, harv, 2975006, A computer program has been created that implements Wagon's spigot algorithm in only 120 characters of software. Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe:{{harvnb|Arndt|Haenel|2006|pp=117, 126â€“128}}JOURNAL, Bailey, David H., David H. Bailey (mathematician), Borwein, Peter B., Peter Borwein, Plouffe, Simon, Simon Plouffe, April 1997, On the Rapid Computation of Various Polylogarithmic Constants, Mathematics of Computation, 66, 218, 903â€“913,weblink 10.1090/S0025-5718-97-00856-9, harv, live,weblink" title="web.archive.org/web/20120722015837weblink">weblink 22 July 2012, dmy-all, 10.1.1.55.3762, 1997MaCom..66..903B,
pi = sum_{k=0}^infty frac{1}{16^k} left( frac{4}{8k + 1} - frac{2}{8k + 4} - frac{1}{8k + 5} - frac{1}{8k + 6}right).

Role and characterizations in mathematics

Because {{pi}} is closely related to the circle, it is found in many formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics, Fourier analysis, and number theory, also include {{pi}} in some of their important formulae.

Geometry and trigonometry

(File:Circle Area.svg|thumb|alt=A diagram of a circle with a square coving the circle's upper right quadrant.|right|The area of the circle equals {{pi}} times the shaded area.){{pi}} appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Below are some of the more common formulae that involve {{pi}}.{{harvnb|BronshteÄ­n|Semendiaev|1971|pp=200, 209}}
• The circumference of a circle with radius {{math|r}} is {{math|2Ï€r}}.
• The area of a circle with radius {{math|r}} is {{math|Ï€r2}}.
• The volume of a sphere with radius {{math|r}} is {{math|{{sfrac|4|3}}Ï€r3}}.
• The surface area of a sphere with radius {{math|r}} is {{math|4Ï€r2}}.
The formulae above are special cases of the volume of the n-dimensional ball and the surface area of its boundary, the (nâˆ’1)-dimensional sphere, given below.Definite integrals that describe circumference, area, or volume of shapes generated by circles typically have values that involve {{pi}}. For example, an integral that specifies half the area of a circle of radius one is given by:{{MathWorld|Semicircle|Semicircle}}
int_{-1}^1 sqrt{1-x^2},dx = frac{pi}{2}.
In that integral the function {{math|{{sqrt|1 âˆ’ x2}}}} represents the top half of a circle (the square root is a consequence of the Pythagorean theorem), and the integral {{math|{{intmath|int|âˆ’1|1}}}} computes the area between that half of a circle and the {{math|x}} axis.File:Sine cosine one period.svg|thumb|340px|alt=Diagram showing graphs of functions|Sine and cosinecosineThe trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement. {{pi}} plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2{{pi}} radians.{{harvnb|Ayers|1964|p=60}} The angle measure of 180Â° is equal to {{pi}} radians, and 1Â° = {{pi}}/180 radians.Common trigonometric functions have periods that are multiples of {{pi}}; for example, sine and cosine have period 2{{pi}},{{harvnb|BronshteÄ­n|Semendiaev|1971|pp=210â€“211}} so for any angle {{math|Î¸}} and any integer {{math|k}},
sintheta = sinleft(theta + 2pi k right) text{ and } costheta = cosleft(theta + 2pi k right).

Eigenvalues

File:Harmonic_partials_on_strings.svg|thumb|right|The overtones of a vibrating string are eigenfunctions of the second derivative, and form a harmonic progression. The associated eigenvalues form the arithmetic progressionarithmetic progressionMany of the appearances of {{pi}} in the formulas of mathematics and the sciences have to do with its close relationship with geometry. However, {{pi}} also appears in many natural situations having apparently nothing to do with geometry.In many applications, it plays a distinguished role as an eigenvalue. For example, an idealized vibrating string can be modelled as the graph of a function {{math|f}} on the unit interval {{math|[0,1]}}, with fixed ends {{math|1=f(0) = f(1) = 0}}. The modes of vibration of the string are solutions of the differential equation {{math|1=f "(x) + Î» f(x) = 0}}. Here {{math|Î»}} is an associated eigenvalue, which is constrained by Sturmâ€“Liouville theory to take on only certain specific values. It must be positive, since the second derivative is negative definite, so it is convenient to write {{math|1=Î» = Î½2}} where {{math|Î½ > 0}} is called the wavenumber. Then {{math|1=f(x) = sin(Ï€ x)}} satisfies the boundary conditions and the differential equation with {{math|1=Î½ = Ï€}}.{{citation |last=Hilbert |first=David |author1link=David Hilbert |last2=Courant |first2=Richard |author2link=Richard Courant |title=Methods of mathematical physics, volume 1 |pages=286â€“290 |year=1966 |publisher=Wiley}}The value {{pi}} is, in fact, the least such value of the wavenumber, and is associated with the fundamental mode of vibration of the string. One way to obtain this is by estimating the energy. The energy satisfies an inequality, Wirtinger's inequality for functions,{{citation|first1=H. |last1=Dym |first2=H.P. |last2=McKean |title=Fourier series and integrals |publisher=Academic Press |year=1972 |page=47}} which states that if a function {{math|f : [0, 1] â†’ â„‚}} is given such that {{math|1=f(0) = f(1) = 0}} and {{math|f}} and {{math|f '}} are both square integrable, then the inequality holds:
pi^2int_0^1|f(x)|^2,dxle int_0^1|f'(x)|^2,dx,
and the case of equality holds precisely when {{math|f}} is a multiple of {{math|sin(Ï€ x)}}. So {{pi}} appears as an optimal constant in Wirtinger's inequality, and from this it follows that it is the smallest such wavenumber, using the variational characterization of the eigenvalue. As a consequence, {{pi}} is the smallest singular value of the derivative on the space of functions on {{math|[0,1]}} vanishing at both endpoints (the Sobolev space H^1_0[0,1]).

Inequalities

File:Sir William Thompson illustration of Carthage.png|thumb|right|The ancient city of Carthage was the solution to an isoperimetric problem, according to a legend recounted by Lord Kelvin {{harv|Thompson|1894}}: those lands bordering the sea that Queen Dido could enclose on all other sides within a single given oxhide, cut into strips.]]The number {{pi}} serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentioned above, it can be characterized via its role as the best constant in the isoperimetric inequality: the area {{mvar|A}} enclosed by a plane Jordan curve of perimeter {{mvar|P}} satisfies the inequality
4pi Ale P^2,
and equality is clearly achieved for the circle, since in that case {{math|1=A = Ï€r{{sup|2}}}} and {{math|1=P = 2Ï€r}}.{{citation|first=Isaac |last=Chavel |title=Isoperimetric inequalities |publisher=Cambridge University Press |year=2001}}Ultimately as a consequence of the isoperimetric inequality, {{pi}} appears in the optimal constant for the critical Sobolev inequality in n dimensions, which thus characterizes the role of {{pi}} in many physical phenomena as well, for example those of classical potential theory.{{citation|author=Talenti, Giorgio|title=Best constant in Sobolev inequality|journal=Annali di Matematica Pura ed Applicata|volume=110|number=1|pages=353â€“372|issn=1618-1891|doi=10.1007/BF02418013|df=dmy-allyear = 1976}}BEST CONSTANTS IN POINCARÃ© INEQUALITIES FOR CONVEX DOMAINS >EPRINT=1110.2960 AUTHOR2=C. NITSCH YEAR=2011 title=Best constants for Gagliardoâ€“Nirenberg inequalities and applications to nonlinear diffusionsauthor2=J. Dolbeaultyear=2002issue=9doi=10.1016/s0021-7824(02)01266-7|citeseerx=10.1.1.57.7077}} In two dimensions, the critical Sobolev inequality is
2pi|f|_2 le |nabla f|_1
for f a smooth function with compact support in {{math|R2}}, nabla f is the gradient of f, and |f|_2 and |nabla f|_1 refer respectively to the {{math|L2}} and {{math|L1}}-norm. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.Wirtinger's inequality also generalizes to higher-dimensional PoincarÃ© inequalities that provide best constants for the Dirichlet energy of an n-dimensional membrane. Specifically, {{pi}} is the greatest constant such that
pi le frac{left (int_G |nabla u|^2right)^{1/2}}{left (int_G|u|^2right)^{1/2}}
for all convex subsets {{math|G}} of {{math|Rn}} of diameter 1, and square-integrable functions u on {{math|G}} of mean zero.{{Citation | last1=Payne | first1=L.E. | last2=Weinberger | first2=H.F. | title=An optimal PoincarÃ© inequality for convex domains | year=1960 | journal=Archive for Rational Mechanics and Analysis | volume=5 | issue=1 | issn=0003-9527 | pages=286â€“292| doi=10.1007/BF00252910 | bibcode=1960ArRMA...5..286P }} Just as Wirtinger's inequality is the variational form of the Dirichlet eigenvalue problem in one dimension, the PoincarÃ© inequality is the variational form of the Neumann eigenvalue problem, in any dimension.

Fourier transform and Heisenberg uncertainty principle

File:Animation_of_Heisenberg_geodesic.gif|thumb|right|An animation of a geodesic in the Heisenberg group, showing the close connection between the Heisenberg group, isoperimetry, and the constant {{pi}}. The cumulative height of the geodesic is equal to the area of the shaded portion of the unit circle, while the arc length (in the Carnotâ€“CarathÃ©odory metricCarnotâ€“CarathÃ©odory metricThe constant {{pi}} also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function {{math|f}} on the real line to the function defined as:
hat{f}(xi) = int_{-infty}^infty f(x) e^{-2pi i xxi},dx.
There are several different conventions for the Fourier transform, all of which involve a factor of {{pi}} that is placed somewhere. The appearance of {{pi}} is essential in these formulas, as there is no possibility to remove {{pi}} altogether from the Fourier transform and its inverse transform. The definition given above is the most canonical, however, because it describes the unique unitary operator on {{math|L{{sup|2}}}} that is also an algebra homomorphism of {{math|L{{sup|1}}}} to {{math|L{{sup|âˆž}}}}.{{citation|title=Harmonic analysis in phase space|author=Gerald Folland|publisher=Princeton University Press|year=1989|page=5|author-link=Gerald Folland}}The Heisenberg uncertainty principle also contains the number {{pi}}. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,
int_{-infty}^infty x^2|f(x)|^2,dx int_{-infty}^infty xi^2|hat{f}(xi)|^2,dxige left(frac{1}{4pi}int_{-infty}^infty |f(x)|^2,dxright)^2.
The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below. The appearance of {{pi}} in the formulae of Fourier analysis is ultimately a consequence of the Stoneâ€“von Neumann theorem, asserting the uniqueness of the SchrÃ¶dinger representation of the Heisenberg group.{{harvnb|Howe|1980}}

Gaussian integrals

File:E^(-x^2).svg|thumb|right|A graph of the 1=Æ’(x) = e{{sup|âˆ’x{{sup|2}}}}}}. The colored region between the function and the x-axis has area {{math|{{sqrt|Ï€}}}}.The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1968, pp. 174â€“190. The Gaussian function, which is the probability density function of the normal distribution with mean {{math|Î¼}} and standard deviation {{math|Ïƒ}}, naturally contains {{pi}}:{{harvnb|BronshteÄ­n|Semendiaev|1971|pp=106â€“107, 744, 748}}
f(x) = {1 over sigmasqrt{2pi} },e^{-(x-mu )^2/(2sigma^2)}.
For this to be a probability density, the area under the graph of {{math|f}} needs to be equal to one. This follows from a change of variables in the Gaussian integral:
int_{-infty}^infty e^{-u^2} , du=sqrt{pi}
which says that the area under the basic bell curve in the figure is equal to the square root of {{pi}}.File:Random walk simulation.png|thumb|right|{{pi}} can be computed from the distribution of zeros of a one-dimensional Wiener processWiener processThe central limit theorem explains the central role of normal distributions, and thus of {{pi}}, in probability and statistics. This theorem is ultimately connected with the spectral characterization of {{pi}} as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function.{{citation|author1=H. Dym|author2=H.P. McKean|title=Fourier series and integrals|publisher=Academic Press|year=1972}}; Section 2.7 Equivalently, {{pi}} is the unique constant making the Gaussian normal distribution {{math|e{{sup|-Ï€x{{sup|2}}}}}} equal to its own Fourier transform.{{citation|author1=Elias Stein|author2=Guido Weiss|title=Fourier analysis on Euclidean spaces|year=1971|publisher=Princeton University Press|page=6|author1-link=Elias Stein}}; Theorem 1.13. Indeed, according to {{harvtxt|Howe|1980}}, the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.

Projective geometry

Let {{math|V}} be the set of all twice differentiable real functions f:mathbb Rtomathbb R that satisfy the ordinary differential equation f(x)+f(x)=0. Then {{math|V}} is a two-dimensional real vector space, with two parameters corresponding to a pair of initial conditions for the differential equation. For any tinmathbb R, let e_t:Vtomathbb R be the evaluation functional, which associates to each fin V the value e_t(f)=f(t) of the function {{math|f}} at the real point {{math|t}}. Then, for each t, the kernel of e_t is a one-dimensional linear subspace of {{math|V''}}. Hence tmapstoker e_t defines a function from mathbb Rtomathbb P(V) from the real line to the real projective line. This function is periodic, and the quantity {{pi}} can be characterized as the period of this map.{{citation|title=Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups|series=Cambridge Tracts in Mathematics|publisher=Cambridge University Press|year=2004|isbn=978-0-521-83186-4|author1=V. Ovsienko|author2=S. Tabachnikov}}: Section 1.3

Topology

File:Order-7 triangular tiling.svg|thumb|right|Uniformization of the Klein quartic, a surface of genus three and Euler characteristic âˆ’4, as a quotient of the hyperbolic plane by the symmetry group PSL(2,7) of the 8Ï€}}, by Gaussâ€“Bonnet.The constant {{pi}} appears in the Gaussâ€“Bonnet formula which relates the differential geometry of surfaces to their topology. Specifically, if a compact surface {{math|Î£}} has Gauss curvature K, then
int_Sigma K,dA = 2pi chi(Sigma)
where {{math|Ï‡(Î£)}} is the Euler characteristic, which is an integer.{{citation|title=A comprehensive introduction to differential geometry|volume=3|author=Michael Spivak|year=1999|publisher=Publish or Perish Press|author-link=Michael Spivak}}; Chapter 6. An example is the surface area of a sphere S of curvature 1 (so that its radius of curvature, which coincides with its radius, is also 1.) The Euler characteristic of a sphere can be computed from its homology groups and is found to be equal to two. Thus we have
A(S) = int_S 1,dA = 2picdot 2 = 4pi
reproducing the formula for the surface area of a sphere of radius 1.The constant appears in many other integral formulae in topology, in particular, those involving characteristic classes via the Chernâ€“Weil homomorphism.{{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = Foundations of Differential Geometry|volume=2| publisher=Wiley Interscience | year=1996|edition=New|page=293|title-link=Foundations of Differential Geometry}}; Chapter XII Characteristic classes

Vector calculus

File:YL10M5sph.png|thumb|right|The techniques of vector calculus can be understood in terms of decompositions into spherical harmonicspherical harmonicVector calculus is a branch of calculus that is concerned with the properties of vector fields, and has many physical applications such as to electricity and magnetism. The Newtonian potential for a point source {{mvar|Q}} situated at the origin of a three-dimensional Cartesian coordinate system isH. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, {{isbn|0-393-96997-5}}.
V(mathbf{x}) = -frac{k Q}{|mathbf{x}|}
which represents the potential energy of a unit mass (or charge) placed a distance {{math|{{abs|x}}}} from the source, and {{mvar|k}} is a dimensional constant. The field, denoted here by {{math|E}}, which may be the (Newtonian) gravitational field or the (Coulomb) electric field, is the negative gradient of the potential:
mathbf{E} = -nabla V.
Special cases include Coulomb's law and Newton's law of universal gravitation. Gauss' law states that the outward flux of the field through any smooth, simple, closed, orientable surface {{mvar|S}} containing the origin is equal to {{math|4{{pi}}kQ}}:
{{oiint|preintegral=4pi k Q = |intsubscpt={scriptstyle S}|integrand=mathbf{E} cdot dmathbf{A}.}}
It is standard to absorb this factor of {{math|4Ï€}} into the constant {{mvar|k}}, but this argument shows why it must appear somewhere. Furthermore, {{math|4Ï€}} is the surface area of the unit sphere, but we have not assumed that {{mvar|S}} is the sphere. However, as a consequence of the divergence theorem, because the region away from the origin is vacuum (source-free) it is only the homology class of the surface {{mvar|S}} in {{math|R3{0} }} that matters in computing the integral, so it can be replaced by any convenient surface in the same homology class, in particular, a sphere, where spherical coordinates can be used to calculate the integral.A consequence of the Gauss law is that the negative Laplacian of the potential {{mvar|V}} is equal to {{math|4Ï€kQ}} times the Dirac delta function:
Delta V(mathbf x) = -4pi k Qdelta(mathbf x).
More general distributions of matter (or charge) are obtained from this by convolution, giving the Poisson equation
Delta V(mathbf x) = -4pi k rho(mathbf x)
where {{math|Ï}} is the distribution function.(File:Spacetime lattice analogy.svg|thumb|right|Einstein's equation states that the curvature of space-time is produced by the matter-energy content.)The constant {{pi}} also plays an analogous role in four-dimensional potentials associated with Einstein's equations, a fundamental formula which forms the basis of the general theory of relativity and describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy:Yeo, Adrian, The pleasures of pi, e and other interesting numbers, World Scientific Pub., 2006, p. 21, {{isbn|978-981-270-078-0}}.Ehlers, JÃ¼rgen, Einstein's Field Equations and Their Physical Implications, Springer, 2000, p. 7, {{isbn|978-3-540-67073-5}}.
R_{munu} - frac{1}{2} R g_{munu} + Lambda g_{munu} = frac{8 pi G}{c^4} T_{munu},
where {{math|R'Î¼Î½}} is the Ricci curvature tensor, {{mvar|R}} is the scalar curvature, {{math|g'Î¼Î½}} is the metric tensor, {{math|Î›}} is the cosmological constant, {{mvar|G}} is Newton's gravitational constant, {{mvar|c}} is the speed of light in vacuum, and {{math|TÎ¼Î½}} is the stressâ€“energy tensor. The left-hand side of Einstein's equation is a non-linear analog of the Laplacian of the metric tensor, and reduces to that in the weak field limit, with the Lambda g term playing the role of a Lagrange multiplier, and the right-hand side is the analog of the distribution function, times {{math|8Ï€}}.

Cauchy's integral formula

(File:Factorial05.jpg|thumb|right|Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles. Here illustrated is the complex logarithm of the Gamma function.)One of the key tools in complex analysis is contour integration of a function over a positively oriented (rectifiable) Jordan curve {{math|Î³}}. A form of Cauchy's integral formula states that if a point {{math|z0}} is interior to {{math|Î³}}, then{{citation|author=Lars Ahlfors|title=Complex analysis|publisher=McGraw-Hill|year=1966|p=115|author-link=Lars Ahlfors}}
oint_gamma frac{dz}{z-z_0} = 2pi i.
Although the curve {{math|Î³}} is not a circle, and hence does not have any obvious connection to the constant {{pi}}, a standard proof of this result uses Morera's theorem, which implies that the integral is invariant under homotopy of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates. More generally, it is true that if a rectifiable closed curve {{math|Î³}} does not contain {{math|z0}}, then the above integral is {{math|2Ï€i}} times the winding number of the curve.The general form of Cauchy's integral formula establishes the relationship between the values of a complex analytic function {{math|f(z)}} on the Jordan curve {{math|Î³}} and the value of {{math|f(z)}} at any interior point {{math|z0}} of {{math|Î³}}:{{mathworld|CauchyIntegralFormula|Cauchy Integral Formula}}Joglekar, S.D., Mathematical Physics, Universities Press, 2005, p. 166, {{isbn|978-81-7371-422-1}}.
oint_gamma { f(z) over z-z_0 },dz = 2pi i f (z_{0})
provided {{math|f(z)}} is analytic in the region enclosed by {{math|Î³}} and extends continuously to {{math|Î³}}. Cauchy's integral formula is a special case of the residue theorem, that if {{math|g(z)}} is a meromorphic function the region enclosed by {{math|Î³}} and is continuous in a neighborhood of {{math|Î³}}, then
oint_gamma g(z), dz =2pi i sum operatorname{Res}( g, a_k )
where the sum is of the residues at the poles of {{math|g(z)}}.

The gamma function and Stirling's approximation

File:Hopf_Fibration.png|thumb|right|The Hopf fibration of the 3-sphere, by Villarceau circles, over the complex projective line with its Fubiniâ€“Study metric (three parallels are shown). The identity {{math|S3(1)/S2(1) {{=}} Ï€/2}} is a consequence.]]The factorial function {{math|n!}} is the product of all of the positive integers through {{math|n}}. The gamma function extends the concept of factorial (normally defined only for non-negative integers) to all complex numbers, except the negative real integers. When the gamma function is evaluated at half-integers, the result contains {{pi}}; for example Gamma(1/2) = sqrt{pi} and Gamma(5/2) = frac {3 sqrt{pi}} {4} .{{harvnb|BronshteÄ­n|Semendiaev|1971|pp=191â€“192}}The gamma function is defined by its Weierstrass product development:{{citation|title =The gamma function|author= Emil Artin|publisher = Holt, Rinehart and Winston|year = 1964|series = Athena series; selected topics in mathematics|edition = 1st|author-link= Emil Artin}}
Gamma(z) = frac{e^{-gamma z}}{z}prod_{n=1}^infty frac{e^{z/n}}{1+z/n}
where {{math|Î³}} is the Eulerâ€“Mascheroni constant. Evaluated at {{math|z {{=}} 1/2}} and squared, the equation {{math|Î“(1/2)2 {{=}} Ï€}} reduces to the Wallis product formula. The gamma function is also connected to the Riemann zeta function and identities for the functional determinant, in which the constant {{pi}} plays an important role.The gamma function is used to calculate the volume {{math|V'n(r)}} of the n-dimensional ball of radius r in Euclidean n-dimensional space, and the surface area {{math|S'nâˆ’1(r)}} of its boundary, the (nâˆ’1)-dimensional sphere:{{citation|author=Lawrence Evans|title=Partial differential equations|publisher=AMS|year=1997|page=615}}.
V_n(r) = frac{pi^{n/2}}{Gammaleft(frac{n}{2}+1right)}r^n,
S_{n-1}(r) = frac{npi^{n/2}}{Gammaleft(frac{n}{2}+1right)}r^{n-1}.
Further, it follows from the functional equation that
2pi r = frac{S_{n+1}(r)}{V_n(r)}.
The gamma function can be used to create a simple approximation to the factorial function {{math|n!}} for large {{math|n}}: n! sim sqrt{2 pi n} left(frac{n}{e}right)^n which is known as Stirling's approximation.{{harvnb|BronshteÄ­n|Semendiaev|1971|p=190}} Equivalently,
pi = lim_{ntoinfty} frac{e^{2n}n!^2}{2 n^{2n+1}}.
As a geometrical application of Stirling's approximation, let {{math|Î”n}} denote the standard simplex in n-dimensional Euclidean space, and {{math|(n + 1)Î”n}} denote the simplex having all of its sides scaled up by a factor of {{math|n + 1}}. Then
operatorname{Vol}((n+1)Delta_n) = frac{(n+1)^n}{n!} sim frac{e^{n+1}}{sqrt{2pi n}}.
Ehrhart's volume conjecture is that this is the (optimal) upper bound on the volume of a convex body containing only one lattice point.{{citation|author1=Benjamin Nill|author2=Andreas Paffenholz|title=On the equality case in Erhart's volume conjecture|year=2014|arxiv=1205.1270|journal=Advances in Geometry|volume=14|issue=4|pages=579â€“586|issn=1615-7168|doi=10.1515/advgeom-2014-0001}}

Number theory and Riemann zeta function

File:PrÃ¼fer.png|thumb|right|Each prime has an associated PrÃ¼fer group, which are arithmetic localizations of the circle. The L-functionL-functionFile:ModularGroup-FundamentalDomain.svg|thumb|right|Solution of the Basel problem using the Weil conjecture: the value of {{math|Î¶(2)}} is the hyperbolic area of a fundamental domain of the 2{{pi}}}}.The Riemann zeta function {{math|Î¶(s)}} is used in many areas of mathematics. When evaluated at {{math|1=s = 2}} it can be written as
zeta(2) = frac{1}{1^2} + frac{1}{2^2} + frac{1}{3^2} + cdots
Finding a simple solution for this infinite series was a famous problem in mathematics called the Basel problem. Leonhard Euler solved it in 1735 when he showed it was equal to {{math|Ï€2/6}}. Euler's result leads to the number theory result that the probability of two random numbers being relatively prime (that is, having no shared factors) is equal to {{math|6/Ï€2}}.{{harvnb|Arndt|Haenel|2006|pp=41â€“43}}This theorem was proved by Ernesto CesÃ ro in 1881. For a more rigorous proof than the intuitive and informal one given here, see Hardy, G.H., An Introduction to the Theory of Numbers, Oxford University Press, 2008, {{isbn|978-0-19-921986-5}}, theorem 332. This probability is based on the observation that the probability that any number is divisible by a prime {{math|p}} is {{math|1/p}} (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is {{math|1/p2}}, and the probability that at least one of them is not is {{math|1 âˆ’ 1/p2}}. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:Ogilvy, C.S.; Anderson, J.T., Excursions in Number Theory, Dover Publications Inc., 1988, pp. 29â€“35, {{isbn|0-486-25778-9}}.
begin{align}
prod_p^infty left(1-frac{1}{p^2}right) &= left( prod_p^infty frac{1}{1-p^{-2}} right)^{-1}[4pt]&= frac{1}{1 + frac{1}{2^2} + frac{1}{3^2} + cdots }[4pt]&= frac{1}{zeta(2)} = frac{6}{pi^2} approx 61%.end{align}This probability can be used in conjunction with a random number generator to approximate {{pi}} using a Monte Carlo approach.{{harvnb|Arndt|Haenel|2006|p=43}}The solution to the Basel problem implies that the geometrically derived quantity {{pi}} is connected in a deep way to the distribution of prime numbers. This is a special case of Weil's conjecture on Tamagawa numbers, which asserts the equality of similar such infinite products of arithmetic quantities, localized at each prime p, and a geometrical quantity: the reciprocal of the volume of a certain locally symmetric space. In the case of the Basel problem, it is the hyperbolic 3-manifold {{math|SL2(R)/SL2(Z)}}.{{citation|title=Algebraic groups and number theory|author1=Vladimir Platonov|author2=Andrei Rapinchuk|publisher=Academic Press|year=1994|pages=262â€“265}}The zeta function also satisfies Riemann's functional equation, which involves {{pi}} as well as the gamma function:
zeta(s) = 2^spi^{s-1} sinleft(frac{pi s}{2}right) Gamma(1-s) zeta(1-s).Furthermore, the derivative of the zeta function satisfies
exp(-zeta'(0)) = sqrt{2pi}.
A consequence is that {{pi}} can be obtained from the functional determinant of the harmonic oscillator. This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula.{{citation|author=Sondow, J.|title=Analytic Continuation of Riemann's Zeta Function and Values at Negative Integers via Euler's Transformation of Series|journal=Proc. Amer. Math. Soc.|volume=120|issue=2|pages=421â€“424|year=1994|doi=10.1090/s0002-9939-1994-1172954-7|citeseerx=10.1.1.352.5774}} The calculation can be recast in quantum mechanics, specifically the variational approach to the spectrum of the hydrogen atom.JOURNAL, 10.1063/1.4930800, T. Friedmann, C.R. Hagen, Quantum mechanical derivation of the Wallis formula for pi, Journal of Mathematical Physics, 56, 11, 112101, 2015, 1510.07813, 2015JMP....56k2101F,

Fourier series

File:2-adic integers with dual colorings.svg|thumb|right|{{pi}} appears in characters of p-adic numbers (shown), which are elements of a PrÃ¼fer group. last1=Tate | first1=John T. | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | publisher=Thompson, Washington, DC | isbn=978-0-9502734-2-6 | mr=0217026 | year=1950 | chapter=Fourier analysis in number fields, and Hecke's zeta-functions | pages=305â€“347}}The constant {{pi}} also appears naturally in Fourier series of periodic functions. Periodic functions are functions on the group {{math|T {{=}}R/Z}} of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued function {{math|f}} on {{math|T}} can be written as an infinite linear superposition of unitary characters of {{math|T}}. That is, continuous group homomorphisms from {{math|T}} to the circle group {{math|U(1)}} of unit modulus complex numbers. It is a theorem that every character of {{math|T}} is one of the complex exponentials e_n(x)= e^{2pi i n x}.There is a unique character on {{math|T}}, up to complex conjugation, that is a group isomorphism. Using the Haar measure on the circle group, the constant {{pi}} is half the magnitude of the Radonâ€“Nikodym derivative of this character. The other characters have derivatives whose magnitudes are positive integral multiples of 2{{pi}}. As a result, the constant {{pi}} is the unique number such that the group T, equipped with its Haar measure, is Pontrjagin dual to the lattice of integral multiples of 2{{pi}}.{{citation|author1=H. Dym|author2=H.P. McKean|title=Fourier series and integrals|publisher=Academic Press|year=1972}}; Chapter 4 This is a version of the one-dimensional Poisson summation formula.

Modular forms and theta functions

File:Lattice_with_tau.svg|thumb|right|Theta functions transform under the lattice of periods of an elliptic curve.]]The constant {{pi}} is connected in a deep way with the theory of modular forms and theta functions. For example, the Chudnovsky algorithm involves in an essential way the j-invariant of an elliptic curve.Modular forms are holomorphic functions in the upper half plane characterized by their transformation properties under the modular group mathrm{SL}_2(mathbb Z) (or its various subgroups), a lattice in the group mathrm{SL}_2(mathbb R). An example is the Jacobi theta function
theta(z,tau) = sum_{n=-infty}^infty e^{2pi i nz + ipi n^2tau}
which is a kind of modular form called a Jacobi form.{{Citation |first=David |last=Mumford |authorlink=David Mumford |title=Tata Lectures on Theta I |year=1983 |publisher=Birkhauser |location=Boston |isbn=978-3-7643-3109-2|pages=1â€“117}} This is sometimes written in terms of the nome q=e^{pi i tau}.The constant {{pi}} is the unique constant making the Jacobi theta function an automorphic form, which means that it transforms in a specific way. Certain identities hold for all automorphic forms. An example is
theta(z+tau,tau) = e^{-pi itau -2pi i z}theta(z,tau),
which implies that {{math|Î¸}} transforms as a representation under the discrete Heisenberg group. General modular forms and other theta functions also involve {{pi}}, once again because of the Stoneâ€“von Neumann theorem.

Cauchy distribution and potential theory

File:Witch of Agnesi, construction.svg|thumb|right|The Witch of Agnesi, named for Maria Agnesi (1718â€“1799), is a geometrical construction of the graph of the Cauchy distribution.]]The Cauchy distribution
g(x)=frac{1}{pi}cdotfrac{1}{x^2+1}
is a probability density function. The total probability is equal to one, owing to the integral:
int_{-infty }^{infty } frac{1}{x^2+1} , dx = pi.
The Shannon entropy of the Cauchy distribution is equal to {{math|ln(4Ï€)}}, which also involves {{pi}}.File:Brownianmotion beads in water spim video.gif|thumb|right|The Cauchy distribution governs the passage of Brownian particles through a membrane.]]The Cauchy distribution plays an important role in potential theory because it is the simplest Furstenberg measure, the classical Poisson kernel associated with a Brownian motion in a half-plane.{{citation|author1=Sidney Port|author2=Charles Stone|title=Brownian motion and classical potential theory|publisher=Academic Press|year=1978|page=29}} Conjugate harmonic functions and so also the Hilbert transform are associated with the asymptotics of the Poisson kernel. The Hilbert transform H is the integral transform given by the Cauchy principal value of the singular integral
Hf(t) = frac{1}{pi}int_{-infty}^infty frac{f(x),dx}{x-t}.
The constant {{pi}} is the unique (positive) normalizing factor such that H defines a linear complex structure on the Hilbert space of square-integrable real-valued functions on the real line.* {{citation|last=Titchmarsh|first=E.|authorlink=Edward Charles Titchmarsh|title=Introduction to the theory of Fourier integrals|isbn=978-0-8284-0324-5|year=1948|edition=2nd|publication-date=1986|publisher=Clarendon Press|location=Oxford University}}. The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space {{math|L2(R)}}: up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line.{{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Singular integrals and differentiability properties of functions|publisher=Princeton University Press|year=1970}}; Chapter II. The constant {{pi}} is the unique normalizing factor that makes this transformation unitary.

Complex dynamics

File:Mandel zoom 00 mandelbrot set.jpg|alt=An complex black shape on a blue background.|thumb|{{pi}} can be computed from the (âˆ’0.75, Îµ)}} diverges.An occurrence of {{pi}} in the Mandelbrot set fractal was discovered by David Boll in 1991.JOURNAL, Klebanoff, Aaron, 2001, Pi in the Mandelbrot set, Fractals, 9, 4, 393â€“402,weblinkweblink" title="web.archive.org/web/20111027155739weblink">weblink 27 October 2011, 14 April 2012, 10.1142/S0218348X01000828, harv, dead, He examined the behavior of the Mandelbrot set near the "neck" at {{nowrap|(âˆ’0.75, 0)}}. If points with coordinates {{nowrap|(âˆ’0.75, Îµ)}} are considered, as Îµ tends to zero, the number of iterations until divergence for the point multiplied by Îµ converges to {{pi}}. The point {{nowrap|(0.25, Îµ)}} at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of Îµ tends to {{pi}}.Peitgen, Heinz-Otto, Chaos and fractals: new frontiers of science, Springer, 2004, pp. 801â€“803, {{isbn|978-0-387-20229-7}}.

Outside mathematics

Describing physical phenomena

Although not a physical constant, {{pi}} appears routinely in equations describing fundamental principles of the universe, often because of {{pi}}'s relationship to the circle and to spherical coordinate systems. A simple formula from the field of classical mechanics gives the approximate period {{math|T}} of a simple pendulum of length {{math|L}}, swinging with a small amplitude ({{math|g}} is the earth's gravitational acceleration):Halliday, David; Resnick, Robert; Walker, Jearl, Fundamentals of Physics, 5th Ed., John Wiley & Sons, 1997, p. 381, {{isbn|0-471-14854-7}}.
T approx 2pi sqrtfrac{L}{g}.
One of the key formulae of quantum mechanics is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Î”{{math|x}}) and momentum (Î”{{math|p}}) cannot both be arbitrarily small at the same time (where {{math|h}} is Planck's constant):WEB, James M., Imamura,weblink Heisenberg Uncertainty Principle, University of Oregon, 17 August 2005, 9 September 2007,weblink" title="web.archive.org/web/20071012060715weblink">weblink 12 October 2007,
Delta x, Delta p ge frac{h}{4pi}.
The fact that {{pi}} is approximately equal to 3 plays a role in the relatively long lifetime of orthopositronium. The inverse lifetime to lowest order in the fine-structure constant {{math|Î±}} isBOOK, Itzykson, C., Claude Itzykson, Zuber, J.-B., Jean-Bernard Zuber, Quantum Field Theory, 1980, Dover Publications, Mineola, NY, 978-0-486-44568-7, 2005,weblink 2005053026, 61200849,
frac{1}{tau} = 2frac{pi^2 - 9}{9pi}malpha^{6},
where {{math|m}} is the mass of the electron.{{pi}} is present in some structural engineering formulae, such as the buckling formula derived by Euler, which gives the maximum axial load {{math|F}} that a long, slender column of length {{math|L}}, modulus of elasticity {{math|E}}, and area moment of inertia {{math|I}} can carry without buckling:Low, Peter, Classical Theory of Structures Based on the Differential Equation, CUP Archive, 1971, pp. 116â€“118, {{isbn|978-0-521-08089-7}}.
F =frac{pi^2EI}{L^2}.
The field of fluid dynamics contains {{pi}} in Stokes' law, which approximates the frictional force {{math|F}} exerted on small, spherical objects of radius {{math|R}}, moving with velocity {{math|v}} in a fluid with dynamic viscosity {{math|Î·}}:Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press, 1967, p. 233, {{isbn|0-521-66396-2}}.
F =6pieta Rv.
In electromagnetics, the vacuum permeability constant Î¼0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation. It is defined as exactly
mu_0 = 4 pi times 10^{-7}text{ H/m} approx 1.2566370614 ldots times 10 ^{-6} text{ N/A}^2.
A relation for the speed of light in vacuum, {{math|c}} can be derived from Maxwell's equations in the medium of classical vacuum using a relationship between Î¼0 and the electric constant (vacuum permittivity), {{math|Îµ0}} in SI units:
c={1oversqrt{mu_0varepsilon_0}}.
Under ideal conditions (uniform gentle slope on a homogeneously erodible substrate), the sinuosity of a meandering river approaches {{pi}}. The sinuosity is the ratio between the actual length and the straight-line distance from source to mouth. Faster currents along the outside edges of a river's bends cause more erosion than along the inside edges, thus pushing the bends even farther out, and increasing the overall loopiness of the river. However, that loopiness eventually causes the river to double back on itself in places and "short-circuit", creating an ox-bow lake in the process. The balance between these two opposing factors leads to an average ratio of {{pi}} between the actual length and the direct distance between source and mouth.JOURNAL, Science (journal), Science, 271, 5256, 22 March 1996, 1710â€“1713, 10.1126/science.271.5256.1710, River Meandering as a Self-Organization Process, Hans-Henrik StÃ¸lum, 1996Sci...271.1710S, harv, {{harvnb|Posamentier|Lehmann|2004|pp=140â€“141}}

In computer culture

In contemporary internet culture, individuals and organizations frequently pay homage to the number {{pi}}. For instance, the computer scientist Donald Knuth let the version numbers of his program TeX approach {{pi}}. The versions are 3, 3.1, 3.14, and so forth.JOURNAL,weblink The Future of TeX and Metafont, Donald, Knuth, Donald Knuth, TeX Mag, 5, 1, 145, 1990-10-03, 2017-02-17, live,weblink" title="web.archive.org/web/20160413230304weblink">weblink 13 April 2016, dmy-all,

References

Footnotes{{notelist}}Citations{{Reflist}}Sources
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• BOOK, Ayers, Frank, Calculus, McGraw-Hill, 1964, 978-0-07-002653-7, harv,
• BOOK, Beckmann, Peter, History of Pi, St. Martin's Press, 1989, 1974, 978-0-88029-418-8, harv,
• BOOK, Berggren, Lennart, Borwein, Jonathan, Jonathan Borwein, Borwein, Peter, Peter Borwein, Pi: a Source Book, Springer-Verlag, 1997, 978-0-387-20571-7, harv,
• BOOK, Borwein, Jonathan, Borwein, Peter, Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity, Wiley, 1987, 978-0-471-31515-5, harv,
• BOOK, Boyer, Carl B., Merzbach, Uta C., Uta Merzbach, 1991, A History of Mathematics, 2, Wiley, 978-0-471-54397-8, harv,
• BOOK, BronshteÄ­n, Ilia, Semendiaev, K.A., A Guide Book to Mathematics, Verlag Harri Deutsch, 1971, 978-3-87144-095-3, harv, A Guide Book to Mathematics,
• BOOK, Eymard, Pierre, Lafon, Jean Pierre, The Number Pi, American Mathematical Society, 1999, 978-0-8218-3246-2, harv, , English translation by Stephen Wilson.
• {{citation|first=Roger|last=Howe|title=On the role of the Heisenberg group in harmonic analysis|journal=Bulletin of the American Mathematical Society|volume=3|pages=821â€“844|number=2|year=1980| doi = 10.1090/S0273-0979-1980-14825-9 | mr = 578375}}.
• BOOK, Joseph, George Gheverghese, The Crest of the Peacock: Non-European Roots of Mathematics, Princeton University Press, 1991, 978-0-691-13526-7,weblink harv, 2013-06-05,
• BOOK, Posamentier, Alfred S., Lehmann, Ingmar, Pi: A Biography of the World's Most Mysterious Number, Prometheus Books, 2004, 978-1-59102-200-8, harv,
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• JOURNAL, Roy, Ranjan, The Discovery of the Series Formula for pi by Leibniz, Gregory, and Nilakantha, Mathematics Magazine, 63, 5, 1990, 291â€“306, 10.2307/2690896, harv, 2690896,
• JOURNAL, Schepler, H.C., The Chronology of Pi, Mathematics Magazine, 1950, 23, 3, 165â€“170 (Jan/Feb), 216â€“228 (Mar/Apr), and 279â€“283 (May/Jun), 10.2307/3029284, harv, 3029284, . issue 3 Jan/Feb, issue 4 Mar/Apr, issue 5 May/Jun
• {{citation|first=William|last=Thompson|authorlink=Lord Kelvin|title=Isoperimetrical problems|year=1894|journal=Nature Series: Popular Lectures and Addresses|volume=II|pages=571â€“592}}

• BOOK, Blatner, David, The Joy of Pi, Walker & Company, 1999, 978-0-8027-7562-7,
• JOURNAL, Borwein, Jonathan, Jonathan Borwein, Peter Borwein, Borwein, Peter, 1984, The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions,weblink SIAM Review, 26, 3, 351â€“365, 10.1137/1026073, 10.1.1.218.8260,
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