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Bessel function
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(File:Vibrating drum Bessel function.gif|thumb|right|Bessel functions are the radial part of the modes of vibration of a circular drum.)Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions {{math|y(x)}} of Bessel's differential equation
x^2 frac{d^2 y}{dx^2} + x frac{dy}{dx} + left(x^2 - alpha^2 right)y = 0
for an arbitrary complex number {{mvar|α}}, the order of the Bessel function. Although {{mvar|α}} and {{math|−α}} produce the same differential equation for real {{mvar|α}}, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of {{mvar|α}}.The most important cases are when {{mvar|α}} is an integer or half-integer.Bessel functions for integer {{mvar|α}} are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer {{mvar|α}} are obtained when the Helmholtz equation is solved in spherical coordinates.

Applications of Bessel functions

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order ({{math|α {{=}} n}}); in spherical problems, one obtains half-integer orders ({{math|α {{=}} n + {{sfrac|1|2}}}}). For example: Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).

Definitions

Because this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.
{| class="wikitable"
! Type !! First kind !! Second kind| Bessel functions
Jα}}Yα}}
#Modified Bessel functions>Modified Bessel functionsIα}}Kα}}
#Hankel functions>Hankel functionsH{{suα>p=(1)}} {{=}} Jα + iYα}}H{{suα>p=(2)}} {{=}} Jα − iYα}}
#Spherical Bessel functions>Spherical Bessel functionsjn}}yn}}
#Spherical Hankel functions>Spherical Hankel functionsh{{sun>p=(1)}} {{=}} jn + iyn}}h{{sun>p=(2)}} {{=}} jn − iyn}}
Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by {{mvar|Nn}} and {{mvar|nn}} respectively, rather than {{mvar|Yn}} and {{mvar|yn}}.{{MathWorld|title=Spherical Bessel Function of the Second Kind|id=SphericalBesselFunctionoftheSecondKind}}{{MathWorld|title=Bessel Function of the Second Kind|id=BesselFunctionoftheSecondKind}}

{{anchor|Bessel functions of the first kind}}Bessel functions of the first kind: {{math|Jα}}

Bessel functions of the first kind, denoted as {{math|Jα(x)}}, are solutions of Bessel's differential equation that are finite at the origin ({{math|x {{=}} 0}}) for integer or positive α and diverge as {{mvar|x}} approaches zero for negative non-integer {{mvar|α}}. It is possible to define the function by its series expansion around {{math|x {{=}} 0}}, which can be found by applying the Frobenius method to Bessel's equation:Abramowitz and Stegun, p. 360, 9.1.10.
J_alpha(x) = sum_{m=0}^infty frac{(-1)^m}{m! , Gamma(m+alpha+1)} {left(frac{x}{2}right)}^{2m+alpha},
where {{math|Γ(z)}} is the gamma function, a shifted generalization of the factorial function to non-integer values. The Bessel function of the first kind is an entire function if {{mvar|α}} is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to {{math|{{frac|1|{{sqrt|x}}}}}} (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large {{mvar|x}}. (The series indicates that {{math|−J1(x)}} is the derivative of {{math|J0(x)}}, much like {{math|−sin x}} is the derivative of {{math|cos x}}; more generally, the derivative of {{math|Jn(x)}} can be expressed in terms of {{math|Jn ± 1(x)}} by the identities below.)thumb|300px|right|Plot of Bessel function of the first kind, {{math|Jα(x)}}, for integer orders {{math|α {{=}} 0, 1, 2}}For non-integer {{mvar|α}}, the functions {{math|Jα(x)}} and {{math|J−α(x)}} are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order {{mvar|α}}, the following relationship is valid (note that the Gamma function has simple poles at each of the non-positive integers):Abramowitz and Stegun, p. 358, 9.1.5.
J_{-n}(x) = (-1)^n J_n(x).
This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

Bessel's integrals

Another definition of the Bessel function, for integer values of {{mvar|n}}, is possible using an integral representation:BOOK, Temme, Nico M., Special Functions: An introduction to the classical functions of mathematical physics, 1996, Wiley, New York, 0471113131, 228–231, 2nd print,
J_n(x) = frac{1}{pi} int_{0}^pi cos (n tau - x sin tau) ,dtau.
Another integral representation is:
J_n(x) = frac{1}{2 pi} int_{-pi}^pi e^{i( x sin tau -n tau)} ,dtau.
This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for {{math|Re(x) > 0}}:Watson, p. 176WEB,weblink Archived copy, 2010-10-18, yes,weblink" title="web.archive.org/web/20100923194031weblink">weblink 2010-09-23, WEB,weblink Integral representations of the Bessel function, www.nbi.dk, 25 March 2018, Arfken & Weber, exercise 11.1.17.
J_alpha(x) = frac{1}{pi} int_0^pi cos(alphatau - x sintau),dtau - frac{sin alphapi}{pi} int_0^infty e^{-x sinh t - alpha t} , dt.

Relation to hypergeometric series

The Bessel functions can be expressed in terms of the generalized hypergeometric series asAbramowitz and Stegun, p. 362, 9.1.69.
J_alpha(x) = frac{left(frac{x}{2}right)^alpha}{Gamma(alpha+1)} ;_0F_1 left(alpha+1; -frac{x^2}{4}right).
This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.

Relation to Laguerre polynomials

In terms of the Laguerre polynomials {{mvar|Lk}} and arbitrarily chosen parameter {{mvar|t}}, the Bessel function can be expressed asBOOK, Gábor Szegő, Szegő, Gábor, Orthogonal Polynomials, 4th, Providence, RI, AMS, 1975,
frac{J_alpha(x)}{left( frac{x}{2}right)^alpha} = frac{e^{-t}}{Gamma(alpha+1)} sum_{k=0}^infty frac{L_k^{(alpha)}left( frac{x^2}{4 t}right)}{binom{k+alpha}{k}} frac{t^k}{k!}.

{{anchor|Weber functions|Neumann functions|Bessel functions of the second kind}}Bessel functions of the second kind: {{math|Yα}}

The Bessel functions of the second kind, denoted by {{math|Yα(x)}}, occasionally denoted instead by {{math|Nα(x)}}, are solutions of the Bessel differential equation that have a singularity at the origin ({{math|x {{=}} 0}}) and are multivalued. These are sometimes called Weber functions, as they were introduced by {{harvs|txt|authorlink=Heinrich Martin Weber|first=H. M.|last=Weber|year=1873}}, and also Neumann functions after Carl Neumannweblinkthumb|300px|right|Plot of Bessel function of the second kind, {{math|Yα(x)}}, for integer orders {{math|α {{=}} 0, 1, 2}}For non-integer {{mvar|α}}, {{math|Yα(x)}} is related to {{math|Jα(x)}} by
Y_alpha(x) = frac{J_alpha(x) cos alphapi - J_{-alpha}(x)}{sin alphapi}.
In the case of integer order {{mvar|n}}, the function is defined by taking the limit as a non-integer {{mvar|α}} tends to {{mvar|n}}:
Y_n(x) = lim_{alpha to n} Y_alpha(x).
If {{mvar|n}} is a nonnegative integer, we have the seriesNIST Digital Library of Mathematical Functions, (10.8.1). Accessed on line Oct. 25, 2016.
begin{align}
Y_n(z) = &-frac{left(frac{z}{2}right)^{-n}}{pi}sum_{0le kle n-1} frac{(n-k-1)!}{k!}left(frac{z^2}{4}right)^k +frac{2}{pi} J_n(z) ln frac{z}{2} [5pt]&-frac{left(frac{z}{2}right)^n}{pi}sum_{kge 0} bigl(psi(k+1)+psi(n+k+1)bigr) frac{left(-frac{z^2}{4}right)^k}{k!(n+k)!}.end{align}where psi(z) is the digamma function, the logarithmic derivative of the gamma function.{{MathWorld |title=Bessel Function of the Second Kind |id=BesselFunctionoftheSecondKind}}There is also a corresponding integral formula (for {{math|Re(x) > 0}}):Watson, p. 178.
Y_n(x) = frac{1}{pi} int_0^pi sin(x sintheta - ntheta) , dtheta -frac{1}{pi} int_0^infty left( e^{n t} + (-1)^n e^{-n t} right) e^{-x sinh t} , dt.{{math|Yα(x)}} is necessary as the second linearly independent solution of the Bessel's equation when {{mvar|α}} is an integer. But {{math|Yα(x)}} has more meaning than that. It can be considered as a "natural" partner of {{math|Jα(x)}}. See also the subsection on Hankel functions below.When {{mvar|α}} is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:
Y_{-n}(x) = (-1)^n Y_n(x).
Both {{math|Jα(x)}} and {{math|Yα(x)}} are holomorphic functions of {{mvar|x}} on the complex plane cut along the negative real axis. When {{mvar|α}} is an integer, the Bessel functions {{mvar|J}} are entire functions of {{mvar|x}}. If {{mvar|x}} is held fixed at a non-zero value, then the Bessel functions are entire functions of {{mvar|α}}.The Bessel functions of the second kind when {{mvar|α}} is an integer is an example of the second kind of solution in Fuchs's theorem.

{{anchor|Hankel functions}}Hankel functions: {{math|H{{su|bα|p(1)}}}}, {{math|H{{su|bα|p(2)}}}}

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, {{math|H{{su|b=α|p=(1)}}(x)}} and {{math|H{{su|b=α|p=(2)}}(x)}}, defined asAbramowitz and Stegun, p. 358, 9.1.3, 9.1.4.
begin{align}
H_alpha^{(1)}(x) &= J_alpha(x) + iY_alpha(x), H_alpha^{(2)}(x) &= J_alpha(x) - iY_alpha(x),end{align}where {{mvar|i}} is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named after Hermann Hankel.The importance of Hankel functions of the first and second kind lies more in theoretical development rather than in application. These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of the factor of the form {{math|eif(x)}}. The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency).Using the previous relationships, they can be expressed as
begin{align}
H_alpha^{(1)}(x) &= frac{J_{-alpha}(x) - e^{-alpha pi i} J_alpha(x)}{i sin alphapi}, H_alpha^{(2)}(x) &= frac{J_{-alpha}(x) - e^{alpha pi i} J_alpha(x)}{- i sin alphapi}.end{align}If {{mvar|α}} is an integer, the limit has to be calculated. The following relationships are valid, whether {{mvar|α}} is an integer or not:Abramowitz and Stegun, p. 358, 9.1.6.
begin{align}
H_{-alpha}^{(1)}(x) &= e^{alpha pi i} H_alpha^{(1)} (x), H_{-alpha}^{(2)}(x) &= e^{-alpha pi i} H_alpha^{(2)} (x).end{align}In particular, if {{math|α {{=}} m + {{sfrac|1|2}}}} with {{mvar|m}} a nonnegative integer, the above relations imply directly that
begin{align}
J_{-(m+frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+frac{1}{2}}(x), Y_{-(m+frac{1}{2})}(x) &= (-1)^m J_{m+frac{1}{2}}(x).end{align}These are useful in developing the spherical Bessel functions (see below).The Hankel functions admit the following integral representations for {{math|Re(x) > 0}}:Abramowitz and Stegun, p. 360, 9.1.25.
begin{align}
H_alpha^{(1)}(x) &= frac{1}{pi i}int_{-infty}^{+infty + ipi} e^{xsinh t - alpha t} , dt, H_alpha^{(2)}(x) &= -frac{1}{pi i}int_{-infty}^{+infty - ipi} e^{xsinh t - alpha t} , dt,end{align}where the integration limits indicate integration along a contour that can be chosen as follows: from {{math|−∞}} to 0 along the negative real axis, from 0 to {{math|±iπ}} along the imaginary axis, and from {{math|±iπ}} to {{math|+∞ ± iπ}} along a contour parallel to the real axis.

{{anchor|Modified Bessel functions|Modified Bessel functions: Iα, Kα}}Modified Bessel functions: {{math|Iα}}, {{math|Kα}}

The Bessel functions are valid even for complex arguments {{mvar|x}}, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined asAbramowitz and Stegun, p. 375, 9.6.2, 9.6.10, 9.6.11.
begin{align}
I_alpha(x) &= i^{-alpha} J_alpha(ix) = sum_{m=0}^infty frac{1}{m!, Gamma(m+alpha+1)}left(frac{x}{2}right)^{2m+alpha}, K_alpha(x) &= frac{pi}{2} frac{I_{-alpha}(x) - I_alpha(x)}{sin alpha pi},end{align}when {{mvar|α}} is not an integer; when {{mvar|α}} is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments {{mvar|x}}. The series expansion for {{math|Iα(x)}} is thus similar to that for {{math|Jα(x)}}, but without the alternating {{math|(−1)m}} factor.If {{math|−π < arg x ≤ {{sfrac|π|2}}}}, {{math|Kα(x)}} can be expressed as a Hankel function of the first kind:
K_alpha(x) = frac{pi}{2} i^{alpha+1} H_alpha^{(1)}(ix),
and if {{math|−{{sfrac|π|2}} < arg x ≤ π}}, it can be expressed as a Hankel function of the second kind:
K_alpha(x) = frac{pi}{2} (-i)^{alpha+1} H_alpha^{(2)}(-ix).
We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if {{math|−π < arg z ≤ {{sfrac|π|2}}}})Abramowitz and Stegun, p. 375, 9.6.3, 9.6.5.:
begin{align}
J_alpha(iz) &= e^{frac{alpha ipi}{2}} I_alpha(z), Y_alpha(iz) &= e^{frac{(alpha+1)ipi}{2}}I_alpha(z) - frac{2}{pi}e^{-frac{alpha ipi}{2}}K_alpha(z).end{align}{{math|Iα(x)}} and {{math|Kα(x)}} are the two linearly independent solutions to the modified Bessel's equation:Abramowitz and Stegun, p. 374, 9.6.1.
x^2 frac{d^2 y}{dx^2} + x frac{dy}{dx} - left(x^2 + alpha^2 right)y = 0.
Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, {{mvar|Iα}} and {{mvar|Kα}} are exponentially growing and decaying functions respectively. Like the ordinary Bessel function {{mvar|Jα}}, the function {{mvar|Iα}} goes to zero at {{math|x {{=}} 0}} for {{math|α > 0}} and is finite at {{math|x {{=}} 0}} for {{math|α {{=}} 0}}. Analogously, {{mvar|Kα}} diverges at {{math|x {{=}} 0}} with the singularity being of logarithmic type.BOOK, Quantum Electrodynamics, Greiner, Walter, Reinhardt, Joachim, 2009, Springer, 72, 978-3-540-87561-1,
{|
α {{=}} 0, 1, 2, 3}}α {{=}} 0, 1, 2, 3}}
Two integral formulas for the modified Bessel functions are (for {{math|Re(x) > 0}}):Watson, p. 181.
begin{align}
I_alpha(x) &= frac{1}{pi}int_0^pi e^{xcos theta} cos alphatheta ,dtheta - frac{sin alphapi}{pi}int_0^infty e^{-xcosh t - alpha t} ,dt, K_alpha(x) &= int_0^infty e^{-xcosh t} cosh alpha t ,dt.end{align}In some calculations in physics, it can be useful to know that the following relation holds:
2,K_0(omega) = int_{-infty}^infty frac{e^{iomega t}}{sqrt{t^2+1}} ,dt.
It can be proven by showing equality to the above integral definition for {{math|K0}}. This is done by integrating a closed curve in the first quadrant of the complex plane.Modified Bessel functions {{math|K1/3}} and {{math|K2/3}} can be represented in terms of rapidly convergent integralsJOURNAL, M. Kh., Khokonov, Cascade Processes of Energy Loss by Emission of Hard Photons, Journal of Experimental and Theoretical Physics, 99, 4, 690–707, 2004, 10.1134/1.1826160, . Derived from formulas sourced to I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).
begin{align}
K_{1/3}(xi) &= sqrt{3} int_0^infty exp left(- xi left(1+frac{4x^2}{3}right) sqrt{1+frac{x^2}{3}} right) ,dx, K_{2/3}(xi) &= frac{1}{sqrt{3}} int_0^infty frac{3+2x^2}{sqrt{1+frac{x^2}{3}}} exp left(- xi left(1+frac{4x^2}{3}right) sqrt{1+frac{x^2}{3}}right) ,dx.end{align}The modified Bessel function of the second kind has also been called by the following names (now rare):
  • Basset function after Alfred Barnard Basset
  • Modified Bessel function of the third kind
  • Modified Hankel functionReferred to as such in: JOURNAL, Teichroew, D., The Mixture of Normal Distributions with Different Variances, The Annals of Mathematical Statistics, 28, 2, 1957, 510–512, 10.1214/aoms/1177706981,
  • Macdonald function after Hector Munro Macdonald

{{anchor|Spherical Bessel functions}}Spherical Bessel functions: {{math|jn}}, {{math|yn}}

thumb|300px|right|Spherical Bessel functions of the first kind, {{math|jn(x)}}, for {{math|n {{=}} 0, 1, 2}}thumb|300px|right|Spherical Bessel functions of the second kind, {{math|yn(x)}}, for {{math|n {{=}} 0, 1, 2}}When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form
x^2 frac{d^2 y}{dx^2} + 2x frac{dy}{dx} + left(x^2 - n(n + 1)right) y = 0.
The two linearly independent solutions to this equation are called the spherical Bessel functions {{mvar|jn}} and {{mvar|yn}}, and are related to the ordinary Bessel functions {{mvar|Jn}} and {{mvar|Yn}} byAbramowitz and Stegun, p. 437, 10.1.1.
begin{align}
j_n(x) &= sqrt{frac{pi}{2x}} J_{n+frac{1}{2}}(x), y_n(x) &= sqrt{frac{pi}{2x}} Y_{n+frac{1}{2}}(x) = (-1)^{n+1} sqrt{frac{pi}{2x}} J_{-n-frac{1}{2}}(x).end{align}{{mvar|yn}} is also denoted {{mvar|nn}} or {{mvar|ηn}}; some authors call these functions the spherical Neumann functions.The spherical Bessel functions can also be written as (Rayleigh's formulas)Abramowitz and Stegun, p. 439, 10.1.25, 10.1.26.
begin{align}
j_n(x) &= (-x)^n left(frac{1}{x}frac{d}{dx}right)^n,frac{sin x}{x}, y_n(x) &= -(-x)^n left(frac{1}{x}frac{d}{dx}right)^n,frac{cos x}{x}.end{align}The first spherical Bessel function {{math|j0(x)}} is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:Abramowitz and Stegun, p. 438, 10.1.11.
begin{align}
j_0(x) &= frac{sin x}{x}. j_1(x) &= frac{sin x}{x^2} - frac{cos x}{x}, j_2(x) &= left(frac{3}{x^2} - 1right) frac{sin x}{x} - frac{3cos x}{x^2}, j_3(x) &= left(frac{15}{x^3} - frac{6}{x}right) frac{sin x}{x} - left(frac{15}{x^2} - 1right) frac{cos x}{x}end{align}andAbramowitz and Stegun, p. 438, 10.1.12.
begin{align}
y_0(x) &= -j_{-1}(x) & &= -frac{cos x}{x}, y_1(x) &= j_{-2}(x) & &= -frac{cos x}{x^2} - frac{sin x}{x}, y_2(x) &= -j_{-3}(x) & &= left(-frac{3}{x^2} + 1right) frac{cos x}{x} - frac{3sin x}{x^2}, y_3(x) &= j_{-4}(x) & &= left(-frac{15}{x^3} + frac{6}{x}right) frac{cos x}{x} - left(frac{15}{x^2} - 1right) frac{sin x}{x}.end{align}

Generating function

The spherical Bessel functions have the generating functionsAbramowitz and Stegun, p. 439, 10.1.39.
begin{align}
frac{1}{z} cos left(sqrt{z^2 - 2zt}right) &= sum_{n=0}^infty frac{t^n}{n!} j_{n-1}(z), frac{1}{z} sin left(sqrt{z^2 + 2zt}right) &= sum_{n=0}^infty frac{(-t)^n}{n!} y_{n-1}(z).end{align}

Differential relations

In the following, {{mvar|fn}} is any of {{mvar|jn}}, {{mvar|yn}}, {{math|h{{su|b=n|p=(1)}}}}, {{math|h{{su|b=n|p=(2)}}}} for {{math|n {{=}} 0, ±1, ±2, ...}}Abramowitz and Stegun, p. 439, 10.1.23, 10.1.24.
begin{align}
left(frac{1}{z}frac{d}{dz}right)^m big(z^{n+1} f_n(z)big) &= z^{n-m+1} f_{n-m}(z), left(frac{1}{z}frac{d}{dz}right)^m big(z^{-n} f_n(z)big) &= (-1)^m z^{-n-m} f_{n+m}(z).end{align}

{{anchor|Spherical Hankel functions}}Spherical Hankel functions: {{math|h{{su|bn|p(1)}}}}, {{math|h{{su|bn|p(2)}}}}

There are also spherical analogues of the Hankel functions:
begin{align}
h_n^{(1)}(x) &= j_n(x) + i y_n(x), h_n^{(2)}(x) &= j_n(x) - i y_n(x).end{align}In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers {{mvar|n}}:
h_n^{(1)}(x) = (-i)^{n+1} frac{e^{ix}}{x} sum_{m=0}^n frac{i^m}{m!,(2x)^m} frac{(n+m)!}{(n-m)!},
and {{math|h{{su|b=n|p=(2)}}}} is the complex-conjugate of this (for real {{mvar|x}}). It follows, for example, that {{math|j0(x) {{=}} {{sfrac|sin x|x}}}} and {{math|y0(x) {{=}} −{{sfrac|cos x|x}}}}, and so on.The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.

{{anchor|Riccati–Bessel functions}}Riccati–Bessel functions: {{math|Sn}}, {{math|Cn}}, {{math|ξn}}, {{math|ζn}}

Riccati–Bessel functions only slightly differ from spherical Bessel functions:
begin{align}
S_n(x) &= x j_n(x) & &= sqrt{frac{pi x}{2}} J_{n+frac{1}{2}}(x), C_n(x) &= -x y_n(x) & &= -sqrt{frac{pi x}{2}} Y_{n+frac{1}{2}}(x), xi_n(x) &= x h_n^{(1)}(x) & &= sqrt{frac{pi x}{2}} H_{n+frac{1}{2}}^{(1)}(x) & &= S_n(x) - iC_n(x), zeta_n(x) &=x h_n^{(2)}(x) & &= sqrt{frac{pi x}{2}} H_{n+frac{1}{2}}^{(2)}(x) & &= S_n(x) + iC_n(x).end{align}They satisfy the differential equation
x^2 frac{d^2 y}{dx^2} + big(x^2 - n(n + 1)big) y = 0.
For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger's equation with hypothetical cylindrical infinite potential barrier.Griffiths. Introduction to Quantum Mechanics, 2nd edition, p. 154. This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g., Du (2004)JOURNAL, Hong, Du, Mie-scattering calculation, Applied Optics, 43, 9, 1951–1956, 2004, 10.1364/ao.43.001951, for recent developments and references.Following Debye (1909), the notation {{mvar|ψn}}, {{mvar|χn}} is sometimes used instead of {{mvar|Sn}}, {{mvar|Cn}}.

Asymptotic forms

The Bessel functions have the following asymptotic forms. For small arguments {{math|0 < z ≪ {{sqrt|α + 1}}}}, one obtains, when {{mvar|α}} is not a negative integer:
J_alpha(z) sim frac{1}{Gamma(alpha+1)} left( frac{z}{2} right)^alpha.
When {{mvar|α}} is a negative integer, we have
J_alpha(z) sim frac{(-1)^{alpha}}{(-alpha)!} left( frac{2}{z} right)^alpha.
For the Bessel function of the second kind we have three cases:
Y_alpha(z) sim begin{cases}
dfrac{2}{pi} left( ln left(dfrac{z}{2} right) + gamma right) & text{if } alpha = 0,
-dfrac{Gamma(alpha)}{pi} left( dfrac{2}{z} right)^alpha + dfrac{1}{Gamma(alpha+1)} left(dfrac{z}{2} right)^alpha cot(alpha pi) & text{if } alpha text{ is not a non-positive integer (one term dominates unless } alpha text{ is imaginary)},
-dfrac{(-1)^alphaGamma(-alpha)}{pi} left( dfrac{z}{2} right)^alpha & text{if } alphatext{ is a negative integer,}
end{cases}where {{mvar|γ}} is the Euler–Mascheroni constant (0.5772...).For large real arguments {{math|z ≫ {{abs|α2 − {{sfrac|1|4}}}}}}, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless {{mvar|α}} is half-integer) because they have zeros all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of {{math|arg z}} one can write an equation containing a term of order {{math|{{abs|z}}−1}}:Abramowitz and Stegun, p. 364, 9.2.1.
begin{align}
J_alpha(z) &= sqrt{frac{2}{pi z}}left(cos left(z-frac{alphapi}{2} - frac{pi}{4}right) + e^{left|operatorname{Im}(z)right|}mathrm{O}left(|z|^{-1}right)right) && text{for } left|arg zright| < pi, Y_alpha(z) &= sqrt{frac{2}{pi z}}left(sin left(z-frac{alphapi}{2} - frac{pi}{4}right) + e^{left|operatorname{Im}(z)right|}mathrm{O}left(|z|^{-1}right)right) && text{for } left|arg zright| < pi.end{align}(For {{math|α {{=}} {{sfrac|1|2}}}} the last terms in these formulas drop out completely; see the spherical Bessel functions above.) Even though these equations are true, better approximations may be available for complex {{mvar|z}}. For example, {{math|J0(z)}} when {{mvar|z}} is near the negative real line is approximated better by
J_0(z) approx sqrt{frac{-2}{pi z}}cos left(z+frac{pi}{4}right)
than by
J_0(z) approx sqrt{frac{2}{pi z}}cos left(z-frac{pi}{4}right).
The asymptotic forms for the Hankel functions are:
begin{align}
H_alpha^{(1)}(z) &sim sqrt{frac{2}{pi z}}e^{ileft(z-frac{alphapi}{2}-frac{pi}{4}right)} &&text{ for } {-pi} < arg z < 2pi, H_alpha^{(2)}(z) &sim sqrt{frac{2}{pi z}}e^{-ileft(z-frac{alphapi}{2}-frac{pi}{4}right)} && text{ for } {-2pi} < arg z < pi.end{align}These can be extended to other values of {{math|arg z}} using equations relating {{math|H{{su|b=α|p=(1)}}(ze'imπ)}} and {{math|H{{su|b=α|p=(2)}}(ze'imπ)}} to {{math|H{{su|b=α|p=(1)}}(z)}} and {{math|H{{su|b=α|p=(2)}}(z)}}.NIST Digital Library of Mathematical Functions, Section 10.11.It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, {{math|Jα(z)}} is not asymptotic to the average of these two asymptotic forms when {{mvar|z}} is negative (because one or the other will not be correct there, depending on the {{math|arg z}} used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for complex (non-real) {{mvar|z}} so long as {{math|{{abs|z}}}} goes to infinity at a constant phase angle {{math|arg z}} (using the square root having positive real part):
begin{align}
J_alpha(z) &sim frac{1}{sqrt{2pi z}} e^{ ileft(z-frac{alphapi}{2}-frac{pi}{4}right)} && text{ for } {-pi} < arg z < 0, J_alpha(z) &sim frac{1}{sqrt{2pi z}} e^{-ileft(z-frac{alphapi}{2}-frac{pi}{4}right)} && text{ for } 0 < arg z < pi, Y_alpha(z) &sim -ifrac{1}{sqrt{2pi z}} e^{ ileft(z-frac{alphapi}{2}-frac{pi}{4}right)} && text{ for } {-pi} < arg z < 0, Y_alpha(z) &sim -ifrac{1}{sqrt{2pi z}} e^{-ileft(z-frac{alphapi}{2}-frac{pi}{4}right)} && text{ for } 0 < arg z < pi.end{align}For the modified Bessel functions, Hankel developed asymptotic expansions as well:Abramowitz and Stegun, p. 377, 9.7.1.Abramowitz and Stegun, p. 378, 9.7.2.
begin{align}
I_alpha(z) &sim frac{e^z}{sqrt{2pi z}} left(1 - frac{4 alpha^2 - 1}{8z} + frac{left(4 alpha^2 - 1right) left(4 alpha^2 - 9right)}{2! (8z)^2} - frac{left(4 alpha^2 - 1right) left(4 alpha^2 - 9right) left(4 alpha^2 - 25right)}{3! (8z)^3} + cdots right) &&text{for }left|arg zright||title-link=Gradshteyn and Ryzhik |chapter=8.411.10.}}This formula is useful especially when working with Fourier transforms.Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by {{mvar|x}}, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:
int_0^1 x J_alphaleft(x u_{alpha,m}right) J_alphaleft(x u_{alpha,n}right) ,dx = frac{delta_{m,n}}{2} left[J_{alpha+1}left(u_{alpha,m}right)right]^2 = frac{delta_{m,n}}{2} left[J_{alpha}'left(u_{alpha,m}right)right]^2
where {{math|α > −1}}, {{math|δ'm,n}} is the Kronecker delta, and {{math|u'α,m}} is the {{mvar|m}}th zero of {{math|Jα(x)}}. This orthogonality relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function is expanded in the basis of the functions {{math|Jα(x uα,m)}} for fixed {{mvar|α}} and varying {{mvar|m}}.An analogous relationship for the spherical Bessel functions follows immediately:
int_0^1 x^2 j_alphaleft(x u_{alpha,m}right) j_alphaleft(x u_{alpha,n}right) ,dx = frac{delta_{m,n}}{2} left[j_{alpha+1}left(u_{alpha,m}right)right]^2
If one defines a boxcar function of {{mvar|x}} that depends on a small parameter {{mvar|ε}} as:
f_varepsilon(x)=varepsilon operatorname{rect}left(frac{x-1}varepsilonright)
(where {{math|rect}} is the rectangle function) then the Hankel transform of it (of any given order {{math|α > −{{sfrac|1|2}}}}), {{math|gε(k)}}, approaches {{math|Jα(k)}} as {{mvar|ε}} approaches zero, for any given {{mvar|k}}. Conversely, the Hankel transform (of the same order) of {{math|gε(k)}} is {{math|fε(x)}}:
int_0^infty k J_alpha(kx) g_varepsilon(k) ,dk = f_varepsilon(x)
which is zero everywhere except near 1. As {{mvar|ε}} approaches zero, the right-hand side approaches {{math|δ(x − 1)}}, where {{mvar|δ}} is the Dirac delta function. This admits the limit (in the distributional sense):
int_0^infty k J_alpha(kx) J_alpha(k) ,dk = delta(x-1)
A change of variables then yields the closure equation:Arfken & Weber, section 11.2
int_0^infty x J_alpha(ux) J_alpha(vx) ,dx = frac{1}{u} delta(u - v)
for {{math|α > −{{sfrac|1|2}}}}. The Hankel transform can express a fairly arbitrary function {{Clarify|reason=This can probably be precisely qualified e.g. square integrable etc.|date=June 2018}}as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:
int_0^infty x^2 j_alpha(ux) j_alpha(vx) ,dx = frac{pi}{2u^2} delta(u - v)
for {{math|α > −1}}.Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:
A_alpha(x) frac{dB_alpha}{dx} - frac{dA_alpha}{dx} B_alpha(x) = frac{C_alpha}{x}
where {{mvar|Aα}} and {{mvar|Bα}} are any two solutions of Bessel's equation, and {{mvar|Cα}} is a constant independent of {{mvar|x}} (which depends on α and on the particular Bessel functions considered). In particular,
J_alpha(x) frac{dY_alpha}{dx} - frac{dJ_alpha}{dx} Y_alpha(x) = frac{2}{pi x}
and
I_alpha(x) frac{dK_alpha}{dx} - frac{dI_alpha}{dx} K_alpha(x) = -frac{1}{x}
for {{math|α > −1}}.For {{math|α > −1}}, the even entire function of genus 1, {{math|x−αJα(x)}}, has only real zeros. Let
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