12π i ∮

|ζ|=z0

F(ζ )

G(ζ )dz=

F (z0)

G ' (z0)

To find the ak coefficients, add up the previous coefficients up to the kth coefficient.

To find partial fraction decompositions of P(x)

Q(x):

1. Ensure that the degree of P(x) is greater than the degree of Q(x).◦ If it is not, do long division to get some fraction that satisfies these

conditions2. Then factor P and Q3. Expand 1

some factor(s) of Q (x)centered at the singularity due to one factor

outside.4. Now expand 1

some factor(s) of Q (x)as a Taylor series centered at the

singularity caused by the factor outside. This is a.◦ hh

5. Now distribute the factor outside and take the part where it has a negative power.

Laurent Series:

an is a constant

Winding Numbers:1. Try to find the zeros of the polynomial algebraically to know where to

look2. Parametrize a function around said area3. Input the parametrization into the original polynomial

4. If the polynomial evaluated by the parametrization goes around zero n times, it has n zeros in the area within the parametrized curve.

5. Try to determine the behavior of the function along the parametrization