Numerically integrate complex valued function of a real variable and a complex variable

Question

I need to numerically integrate a complex valued function of a real variable and a complex variable by the real variable. My integral:

$$ \int^1_{-1}\frac{e^{ix}}{x-z}dx ~~~\text{where} ~~x \in \mathbb{R}, z \in \mathbb{C}$$

I'm trying to do this:

z := x + I y; NIntegrate[Exp[I x]/(x - z), {x, -1, 1}]

But I get the message:

NIntegrate::inumr: The integrand (I E^(I x))/y has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,1}}.

How can this integration be done numerically in Mathematica?

Answer 1

I will assume that $z = a+ib$, since otherwise you are assuming that $x-z=iy$ which is always imaginary in your question. However, you do not even need to actually specify $z$ in that form. You just need to do this:

F[z_?NumberQ] := Integrate[Exp[I*x]/(x - z), {x, -1, 1}, PrincipalValue -> True]
F[1 + 2 I]
F[-I/2]
F[I/3]
F[0]

This gives

\begin{align} &e^{-2+i} (\text{Ei}(-2)-\text{Ei}(2-2 i))\\ &\sqrt{e} \left(-\text{Ei}\left(-\frac{1}{2}-i\right)+\text{Ei}\left(-\frac{1}{2}+i\right)-2 i \pi \right)\\ &\frac{\text{Ei}\left(\frac{1}{3}+i\right)-\text{Ei}\left(\frac{1}{3}-i\right)}{\sqrt[3]{e}}\\ &2 i \text{Si}(1) \end{align}

The last example is meant to show how it works when $z$ is real number, in which you need principal value integral. You could also change this to numerical case

G[z_?NumberQ] := 
 NIntegrate[Exp[I*x]/(x - z), {x, -1, z, 1}, Method -> PrincipalValue]

where you can account for when $z\in (-1,1)$ and you invoke PrincipalValue as "Method" (slightly different syntax than the Integrate command).

Answer 2

z := x + I*y;

Integrate[Exp[I*x]/(x - z), {x, -1, 1}]

(2 I Sin[1])/y

If you insist on using NIntegrate then you should provide numerical value to y?