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)`