So, here's what I understand:

-A complex number can be seen as a vector. Thus a complex-valued function is taking the "standard" vector field to a different vector field.

-We evaluate an integral of a vector field using line integrals, which use both "x and y" or real and imaginary directions to compute. To do this, we parametrize the curve so that it had a common variable. When we integrate along this curve that is created through parametrization, we are calculating the length of that curve based on the real and imaginary part of the complex number.

What I don't know:

-How is an integral evaluated along one path equivalent to the integral evaluated along another path?

-How do you know what to parametrize the curve to?

-Anything else I am missing?

Background:

Junior undergraduate math major. I've had multivariable calculus with a teacher who couldn't properly teach visualization of the topics. I am in real analysis currently and am self-studying complex analysis for an Independent Study course next semester. I've read the material in the book, but whenever integration is brought up, it is confusing at most parts.

Also, please correct me if there's a better/more right way of thinking about what I said.

Thank you in advance.