Complex Integration

chill_factor

1. The problem statement, all variables and given/known data

Is the integral ∫z* dz from the point (0,0) to (3,2) on the complex plane path dependent?

2. Relevant equations

I = ∫ f(z)dz = ∫udx - vdy + i ∫ vdx + udy

z = x-iy, u = x, v = -y

3. The attempt at a solution

I have no idea how to start. The methods given in the book and from real line integrals don't seem to apply here. For example the book recommends, for real line integrals, to substitute y = x so that it reduces to a single integral. For complex integrals, it is recommended to parameterize f(z) into a f(z(t)) and reduce it to a single integration.

I've tried z = re^iθ, so dz = r*i*e^iθ dθ + e^iθ dr, now how do I reduce this to a single integration?

vela

You can't reduce it to a integration with respect to some parameter t until you choose a path between the two points.

chill_factor

Ok, thank you. I think now I figured the strategy out. Instead of trying to prove it in general that it depends on r, it could be better to select 2 arbitrary paths that are easy to compute: a straight line from 0,0 to the point, and the piecewise path of 2 line segments parallel to their respective axes.

clamtrox

In this case, that is sufficient: as you correctly guessed, the integral is path-dependent. You might want to wonder why this is the case. Cauchy integral theorem will be helpful here.