Help with an integral involving complex number

Question

I have an integral expression involving complex variable $z(x)=|x|^3+i z_i(x)$: $$\int_{-\infty}^\infty e ^{z}dx=2 \text{Re}\int_0^\infty e^{z^+}dx, $$ where $z^+=x^3+i z_i$ and $z(-x)=\overline{z(x)}$. Note that $z$ is not analytic in the complex $x$ plane. The usefulness of the expression is that it converts the integral of a non-analytic function to that of an analytic one in the complex $x$ plane. I would appreciate any help in explaining the equality.

Answer 1

It's a special case of$$\int_{-\infty}^\infty f(x)dx=\int_0^\infty f(x)dx+\int_{-\infty}^0 f(x)dx=\int_0^\infty(f(x)+f(-x))dx,$$so in particular$$\int_{-\infty}^\infty g(|x|)dx=2\int_0^\infty g(x)dx.$$The real part of$$\int_{-\infty}^\infty\exp(|x|^3+iz_i)dx=\int_{-\infty}^\infty\exp|x|^3\cos z_idx+i\int_{-\infty}^\infty\exp|x|^3\sin z_idx.$$transforms as required (although it diverges). The imaginary part does too, because $z_i$ is odd, so$$\int_{-\infty}^\infty\exp|x|^3\sin z_idx=\int_0^\infty\exp(x^3)\underbrace{(\sin z_i(x)+\sin z_i(-x))}_{0}dx=0.$$