Integrate on a contour in the complex plane

Question

I want to integrate the function

f[z]= z + Conjugate[z]

over a circle of radius 2 centered at the origin.

For the sake of stating something that I have tried:

Integrate[z + Conjugate[z], {z, 1, I, -1, -I, 1}] 

seems to give me the integral around the square with vertices 1, I, -1, -I. Also I can get the integral on a line segment with:

Integrate[z + Conjugate[z], {z, 0, 1 + I}].

How do I integrate on a curve?

Answer 1

First we shall define 'the integration on a curve'. Traditionally, this is defined as integration of f.dl where dl is the length of a small part of the curve.

So, using t as a medium, we can explicitly write out the curve's function on a complex plane, here let's assume it's z=2 Exp[I t].

Then we can use t, a real number, as the integration variable, which make this problem significantly easier.

The final code is shown below:

z=2 Exp[I t];
f[z_]:=z+Conjugate[z];
Integrate[f[z] Abs[D[z,t]],{t,0,1}]

result is:

8 Sin[1] 

Hope this answer is helpful to you:)

Answer 2

EDIT

It can get more complicated. Look at a contour given by

r[t_] := 1 + 2 Cos[t];

Original post

The result of the integral for a (reasonable) closed contour is just 2 I times the area enclosed by the contour.

Proof: the integral is

$$\int \left(z^*+z\right) \, dz$$

Letting

$$z=x+i y$$,
$$dz=dx+i dy$$

the integral becomes

$$\int 2 x (dx + i dy)=2 i \int dy* x - 2 \int dx *x $$

It can be easily shown that the second summand vanishes over a closed curve which completes the proof. QED.