Author Topic: 2.5 Example 5  (Read 1298 times)

Jeffery Mcbride

  • Full Member
  • ***
  • Posts: 24
  • Karma: 19
    • View Profile
2.5 Example 5
« on: November 18, 2018, 01:04:15 AM »
Can someone please help me with example 5? I understand the Res(r ; -2i) is F(z)/G'(z) but when I try that I am not getting the same calculation as the book.

**Solved, had to multiply by the conjugate to get the answer.
« Last Edit: November 18, 2018, 01:13:35 AM by Jeff Mcbride »

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 2607
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: 2.5 Example 5
« Reply #1 on: November 18, 2018, 04:38:20 AM »
So, we need to calculate residue in each pole of
Points $\pm 2i$ are easy since they are simple poles, with the residues equal to
\frac{z+1}{(z^2+4)'(z-1)^3}\Bigr|_{z=\pm 2i}=\frac{z+1}{2z (z-1)^3}\Bigr|_{z=\pm 2i}=\frac{\pm 2i+1}{\pm 4i (\pm 2i-1)^3}=...
Point $z=1$ is more tricky since it is a triple pole but the factor $(z-1)^3$ is already separated, so we need to find a coefficient at $(z-1)^2$ in the decomposition of $g(z)=\frac{z+1}{(z^2+4)}$ at $z=1$; it is $\frac{1}{2}g''(z)$ at that point.

But there is a simpler way to find a residue at $z=1$. The function is meromorphic in the extended complex plane, having only isolated points. Then the sum of all residues should be $0$, and we need to include $\infty $ in the tally. But since at infinity the function decays faster than $z^{-1}$, the residue there is $0$. So
\Res (f, 1)= -\Res (f,2i) -\Res (f, -2i).