Exponential integral

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.
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 ∞ {\displaystyle \infty } . Instead of Ei, the following notation is used,
For positive values of x, we have − E 1 ( x ) = Ei ⁡ ( − x ) {\displaystyle -E_{1}(x)=\operatorname {Ei} (-x)} .
In general, a branch cut is taken on the negative real axis and E1 can be defined by analytic continuation elsewhere on the complex plane.
For positive values of the real part of z {\displaystyle z} , this can be written
The behaviour of E1 near the branch cut can be seen by the following relation:
Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.
For real or complex arguments off the negative real axis, E 1 ( z ) {\displaystyle E_{1}(z)} can be expressed as
where γ {\displaystyle \gamma } is the Euler–Mascheroni constant. The sum converges for all complex z {\displaystyle 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 ) {\displaystyle E_{1}(x)} with floating point operations for real x {\displaystyle x} between 0 and 2.5. For x > 2.5 {\displaystyle x>2.5} , the result is inaccurate due to cancellation.
A faster converging series was found by Ramanujan:
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 ) {\displaystyle E_{1}(10)} . However, for positive values of x, there is a divergent series approximation that can be obtained by integrating x e x E 1 ( x ) {\displaystyle xe^{x}E_{1}(x)} by parts:
The relative error of the approximation above is plotted on the figure to the right for various values of N {\displaystyle N} , the number of terms in the truncated sum ( N = 1 {\displaystyle N=1} in red, N = 5 {\displaystyle N=5} in pink).
Using integration by parts, we can obtain an explicit formula Ei ⁡ ( z ) = e z z ( ∑ k = 0 n k ! z k + e n ( z ) ) , e n ( z ) ≡ ( n + 1 ) ! z e − z ∫ − ∞ z e t t n + 2 d t {\displaystyle \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}}}\,dt} For any fixed z {\displaystyle z} , the absolute value of the error term | e n ( z ) | {\displaystyle |e_{n}(z)|} decreases, then increases. The minimum occurs at n ∼ | z | {\displaystyle n\sim |z|} , at which point | e n ( z ) | ≤ 2 π | z | e − | z | {\displaystyle \vert e_{n}(z)\vert \leq {\sqrt {\frac {2\pi }{\vert z\vert }}}e^{-\vert z\vert }} . This bound is said to be "asymptotics beyond all orders".
From the two series suggested in previous subsections, it follows that E 1 {\displaystyle 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 {\displaystyle E_{1}} can be bracketed by elementary functions as follows:
The left-hand side of this inequality is shown in the graph to the left in blue; the central part E 1 ( x ) {\displaystyle E_{1}(x)} is shown in black and the right-hand side is shown in red.
Both Ei {\displaystyle \operatorname {Ei} } and E 1 {\displaystyle E_{1}} can be written more simply using the entire function Ein {\displaystyle \operatorname {Ein} } defined as
(note that this is just the alternating series in the above definition of E 1 {\displaystyle E_{1}} ). Then we have
The function Ein {\displaystyle \operatorname {Ein} } is related to the exponential generating function of the harmonic numbers:
Kummer's equation
is usually solved by the confluent hypergeometric functions M ( a , b , z ) {\displaystyle M(a,b,z)} and U ( a , b , z ) . {\displaystyle U(a,b,z).} But when a = 0 {\displaystyle a=0} and b = 1 , {\displaystyle b=1,} that is,
we have
for all z. A second solution is then given by E1(−z). In fact,
with the derivative evaluated at a = 0. {\displaystyle a=0.} Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z):
The exponential integral is closely related to the logarithmic integral function li(x) by the formula
for non-zero real values of x {\displaystyle x} .
The exponential integral may also be generalized to
which can be written as a special case of the upper incomplete gamma function:
The generalized form is sometimes called the Misra function φ m ( x ) {\displaystyle \varphi _{m}(x)} , defined as
Many properties of this generalized form can be found in the NIST Digital Library of Mathematical Functions.
Including a logarithm defines the generalized integro-exponential function
The derivatives of the generalised functions E n {\displaystyle E_{n}} can be calculated by means of the formula
Note that the function E 0 {\displaystyle E_{0}} is easy to evaluate (making this recursion useful), since it is just e − z / z {\displaystyle e^{-z}/z} .
If z {\displaystyle z} is imaginary, it has a nonnegative real part, so we can use the formula
to get a relation with the trigonometric integrals Si {\displaystyle \operatorname {Si} } and Ci {\displaystyle \operatorname {Ci} } :
The real and imaginary parts of E 1 ( i x ) {\displaystyle \mathrm {E} _{1}(ix)} are plotted in the figure to the right with black and red curves.
There have been a number of approximations for the exponential integral function. These include:
We can express the Inverse function of the exponential integral in power series form:
where μ {\displaystyle \mu } is the Ramanujan–Soldner constant and ( P n ) {\displaystyle (P_{n})} is polynomial sequence defined by the following recurrence relation:
For n > 0 {\displaystyle n>0} , deg ⁡ P n = n {\displaystyle \deg P_{n}=n} and we have the formula :