Finding the Suitable Parametrization for Computing ∫Cr (z - z0)n dz

[nickolas2730]

compute the integral ∫_Cr (z - z₀)ⁿ dz,
with an integer and C_r the circle │z - z₀│= r traversed once in the counterclockwise direction

Solution:

A suitable parametrization for Cr is give by z(t)= z0 + re^it 0≤t≤2π
...
...

My question is , how to find that suitable z(t)?
i have no idea how to find out the z(t), it just pop out in the solution.

Thanks

 

[susskind_leon]

It's a circle in the real-imaginary plane.
A circle in the xy-plane can be parametrized by
x=r*cos(t)
y=r*sin(t)
with 0<=t<=2pi
This comes down to the fact that cos^2(t)+sin^(t)=1 as a circle is defined by x^2+y^2=R^2.
Now bear in mind Euler's formula
e^{ix}=cos(x)+i sin(x)
Alright?

 

[nickolas2730]

so if C is the circle of │z - 2i │= 4
z(t) = 2i + 4e^it ?
i am doing it right?

 

[susskind_leon]

You got it ;)

 

[nickolas2730]

thanks!