1. Presented By Osman Toufiq Ist Year Ist SEM
RESIDUE THEOREM
Presented By
Osman Toufiq
Ist Year Ist SEM

2. RESIDUE THEOREM Calculus of residues
Suppose an analytic function f (z) has an isolated singularity at z0. Consider a contour integral enclosing z0 . z0
The coefficient a-1=Res f (z0) in the Laurent expansion is called the residue of f (z) at z = z0.
If the contour encloses multiple isolated singularities, we have the residue theorem: z0 z1
Contour integral =2pi ×Sum of the residues at the enclosed singular points

3. Residue formula:
To find a residue, we need to do the Laurent expansion and pick up the coefficient a-1. However, in many cases we have a useful residue formula:

6. Residue at infinity:
Stereographic projection:
Suppose f (z) has only isolated singularities, then its residue at infinity is defined as
Another way to prove it is to use Cauchy’s integral theorem. The contour integral for a small loop in an analytic region is
One other equivalent way to calculate the residue at infinity is
By this definition a function may be analytic at infinity but still has a residue there.

7. C
r=1 z+ z- C
r=1

8. C
r=1 z+ z-

9. C
r=1 z+ z- z0

10. Question: How about
Answer: We can go the lower half of the complex plane using a clockwise contour.