Is the residue of a complex function always real?

Question

Let $f(z) = \frac{H(z)}{(z-z_o)^m}$, where H(z) is an analytic function, the residue of f(z) is $Res(f,z_o) = c_{m-1}$

But recall that the residue formulas (integral form) has a $2\pi i$ factor, so it is quite surprising that in most examples I find that the residue is a real number

So I wonder if the residue can be referred to as the "real part" of the principal part of the Laurent series of a complex function.

Can someone help me validate whether the above claim is true?

If you multiply $f$ by a constant $c\in\mathbb{C}$, how are the residues of $f$ and $c\cdot f$ related? — Daniel Fischer, Jan 16 '15 at 01:36
For instance, the residue of $2(z-5)^3+(7-i)+\frac{(3+4i)}{z-5} + \frac{11}{(z-5)^2}$ at $z=5$ is $(3+4i)$ — orangeskid, Jan 16 '15 at 01:59

score 3 · Accepted Answer · answered Jan 16 '15 at 01:38

3

The residue of a Laurent series $f(z) = \sum_{n \in \mathbb{Z}} c_n z^n$ is the coefficient on its $z^{-1}$ term. If we have a function with real residue, i.e. $c_{-1} \in \mathbb{R}$, how does the residue of e.g. the function $i f(z)$ compare?

answered Jan 16 '15 at 01:38

Michael Cromer

2,579

Valentin · Answer 2 · 2022-01-09T09:27:59.313

1

What? No, of course not. Example in comments is quite bad! Here is a better one: Residue of order 3 -

Yes, it has a residue of -3i/512 in a pole in 2i. Cool for a perfect function like:

$$f(z) = \frac{1}{(z^2+4)^3}$$

Another example is $$ \frac{1}{z^3(z^{10}-2)}$$ from Evaluate $\int_{|z|=3}\frac{dz}{z^3(z^{10}-2)}$

edited Jan 09 '22 at 09:27

answered Jan 09 '22 at 08:51

Valentin

46
2

Did you mean $\frac1{z^3(z^{10}-2)}$? – José Carlos Santos Jan 09 '22 at 09:13
Yeah. Corrected. – Valentin Jan 09 '22 at 09:18

Is the residue of a complex function always real?

2 Answers2