The Exponential Integral

Not to be confused with the Exponential Anti-derivative (ie., the integral of exp(z)), the exponential integral, $Ei(z)$ , cannot be evaluated in closed form (with finite terms of elementary functions) and, as such, it is a special function. It is surprising when considering the apparent simplicity of the integrands, but the integration does not lead to a trivial solution. Like other special functions, the exponential integral can be evaluated by approximation to an arbitrary accuracy using infinite series. For a more detailed description see Wolfram functions.

Definition

The Exponential Integral is defined by the integral,

$"E"_1(x)=-int exp(-x)/x dx$

Series Expansion

The function can be approximation using the series expansion, which is useful for evaluation in software,

$E_1(z)=-gamma-lnz-sum_(n=1)^oo (-1)^nz^n/(n!n)$

Hypergeometric Form

It is possible to express the Exponential integral as a hypergeometric function,

$E_1(z)=-gamma-lnz+" "_2F_2(1,1;2,2; -z)z$

Graph

The Polynomial Denominator

Another useful variant is the integral with a square denominator

$int exp(-x)/x^2dx$

We can go about evaluating this by finding the exponential integral by integration by parts. Expressing the exponential integral (with $x^1$ in the denominator) in the form,

$int exp(-x)/xdx=intu'vdx=uv-intv'udx$

We have, $u'=exp(-x)$ and $v=1/x$ , and so $u=-exp(x)$ and $v'=-1/x^2$ . Hence,

$int exp(-x)/x dx = -exp(-x)/x - int exp(-x)/x^2$

Now we recognise the integral on the left as being the definition $-"E"_1(x)$ , and hence we can solve it for the integral on the right. ie.,

$int exp(-x)/x^2 dx = "E"_1(x)-exp(-x)/x$

The Double Integral

Another property which can be useful is the integral of the exponential integral,

$int int exp(-x)/x dx dx = int "E"_1(x) dx=x "E"_1(x)-exp(-x)$