Can you use complex intervals for definite integrals?

Question

Can there be integrals of the form $\int^{a+bi}_{c+di}f(x)\,dx$? I know that definite integrals use intervals on the real number line, but what if I am integrating complex functions? I am new to contour integration and I need help.

Answer 1

In general, the integral between two points in the complex plane is path-dependent. Generally we denote a path $\gamma,$ i.e. $\gamma:[a,b] \to \mathbb C$ and we write $$ \int_\gamma f(z)dz$$ for the complex integral. This can be written in terms of the complex-valued integral over the reals as $$\int_a^b f(\gamma(t))\gamma'(t)dt.$$

Sometimes interesting complex integrals are provably path-independent in which case a notation $$ \int_{z_1}^{z_2} f(z)dz$$ like the one you wrote could be used.

Also, it's common to see integrals written like $$ \int_{a-i\infty}^{a+i\infty} f(z)dz$$ which in context means the integral is over the vertical line.