# Parametric definition for a complex integral

1. ### thayin

4
I have been working in complex functions and this is a new animal I came across.

Let γ be a piecewise smooth curve from -1 to 1, and let
A=∫γa(x2-y2) + 2bxy dz
B=∫γ2axy - b(x2-y2) dz

Prove A + Bi = (2/3)(a-bi)

In the past anything like this defined γ and I would find a parametric definition of the function γ and integrate. This seems like a completely different animal.
Does anyone have any ideas as to how to bet this one rolling?

2. ### mathwonk

9,814
there are certain integrands called "exact" that give the same answer over every path joining the two endpoints. These are the ones which equal df for some function f. maybe this is one. do you know the test for exactness

or maybe you can just guess f. have you ever multiplied out z^2 = (x+iy)^2?

If not you need to practice many more basic examples of complex arithmetic.