Coursera Complex Analysis Lesson 7.4 Finding Residues
Aug 19, 2016
IMPORTANT
Slide 2 Restate Residue Theorem
- f has isolated singularity at
and is analytic on the punctured disc 0 < | z-
| < r
- f has a unique Laurent series representation …
- The residue of f at
is represented by Res(f,
) =
the coefficient of
Following slides are to calculate residues at removable singularities and poles (not essential singularities?)
Slide 3 Residue at Removable singularity
is a removable singularity of a Laurent series if there are no negative powers of the terms
so = 0
so Res(f,) = 0
Slide 4 Residue at simple pole
(*)
How to isolate ? Cross-multiply then take limit as z tends to
We need to take limit because of the division by in the original Laurent series (*)
Slide 5 Residue at simple pole example
(From previous slide)
Consider has simple poles at
= +i, -i
Slide 6 Residue at double pole example
Laurent series: (*)
is double pole when
for k > 2.
To isolate . (i) Multiply by (z – z_0)^2 and (ii) differentiate (iii) take limit.
Slide 7 Residue at double pole example
double pole at z = 1, simple pole at z = 3.
Slide 8 Residue at poles of order n
Use method above: (i) Multiply by (z – z_0)^n and (ii) differentiate multiple times (iii) take limit.
Giving
Res(f,z_0) =
Slide 9 Residue of “quotient” functions
f(z) =
Res(f,) =