{{Use American English|date = January 2019}}{{Short description|Special function defined by an integral}}{{distinguish|text=other integrals of exponential functions}}(File:Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D)In mathematics, the exponential integral Ei is a special function on the complex plane.It is defined as one particular definite integral of the ratio between an exponential function and its argument.

Definitions

For real non-zero values of x, the exponential integral Ei(x) is defined as The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero.For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and {{nowrap|infty.}}Abramowitz and Stegun, p. 228 Instead of Ei, the following notation is used,Abramowitz and Stegun, p. 228, 5.1.1

Properties

Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.

Convergent series

300px|right|thumb| Plot of E_1 function (top) and operatorname{Ei} function (bottom).For real or complex arguments off the negative real axis, E_1(z) can be expressed asAbramowitz and Stegun, p. 229, 5.1.11 where gamma is the Euler–Mascheroni constant. The sum converges for all complex z, and we take the usual value of the complex logarithm having a branch cut along the negative real axis.This formula can be used to compute E_1(x) with floating point operations for real x between 0 and 2.5. For x > 2.5, the result is inaccurate due to cancellation.A faster converging series was found by Ramanujan:

Asymptotic (divergent) series

Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, more than 40 terms are required to get an answer correct to three significant figures for E_1(10).Bleistein and Handelsman, p. 2 However, for positive values of x, there is a divergent series approximation that can be obtained by integrating x e^x E_1(x) by parts:Bleistein and Handelsman, p. 3 The relative error of the approximation above is plotted on the figure to the right for various values of N, the number of terms in the truncated sum (N=1 in red, N=5 in pink).

Asymptotics beyond all orders

Using integration by parts, we can obtain an explicit formula{{Citation |last=O’Malley |first=Robert E. |title=Asymptotic Approximations |date=2014 |url=https://doi.org/10.1007/978-3-319-11924-3_2 |work=Historical Developments in Singular Perturbations |pages=27–51 |editor-last=O'Malley |editor-first=Robert E. |access-date=2023-05-04 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-11924-3_2 |isbn=978-3-319-11924-3}}operatorname{Ei}(z) = frac{e^{z}} {z} left (sum _{k=0}^{n} frac{k!} {z^{k}} + e_{n}(z)right), quad e_{n}(z) equiv (n + 1)! ze^{-z}int _{ -infty }^{z} frac{e^{t}} {t^{n+2}},dtFor any fixed z, the absolute value of the error term |e_n(z)| decreases, then increases. The minimum occurs at nsim |z|, at which point vert e_{n}(z)vert leq sqrt{frac{2pi } {vert zvert }}e^{-vert zvert }. This bound is said to be "asymptotics beyond all orders".

Exponential and logarithmic behavior: bracketing

From the two series suggested in previous subsections, it follows that E_1 behaves like a negative exponential for large values of the argument and like a logarithm for small values. For positive real values of the argument, E_1 can be bracketed by elementary functions as follows:Abramowitz and Stegun, p. 229, 5.1.20 frac 1 2 e^{-x},ln!left( 1+frac 2 x right)< E_1(x) < e^{-x},ln!left( 1+frac 1 x right)qquad x>0The left-hand side of this inequality is shown in the graph to the left in blue; the central part E_1(x) is shown in black and the right-hand side is shown in red.

Definition by Ein

Both operatorname{Ei} and E_1 can be written more simply using the entire function operatorname{Ein}Abramowitz and Stegun, p. 228, see footnote 3. defined as operatorname{Ein}(z)

int_0^z (1-e^{-t})frac{dt}{t}

sum_{k1}^infty frac{(-1)^{k+1}z^k}{k; k!}

(note that this is just the alternating series in the above definition of E_1). Then we have E_1(z) ,=, -gamma-ln z + {rm Ein}(z)qquad left| operatorname{Arg}(z) right| < pi qquad x neq 0

Relation with other functions

Kummer's equation is usually solved by the confluent hypergeometric functions M(a,b,z) and U(a,b,z). But when a=0 and b=1, that is, we have for all z. A second solution is then given by E1(−z). In fact,