Complex numbers - residue theorem

[hermanni]

Hi all,
I'm trying to solve 4.15 from the attached file, can anyone help? I tried to use residue thm , i.e the integral of f over the curve gamma-r equals winding number of z0 over gamma-r and residue of z0 of f. I can't see how b-a relates to the winding number of z0. Can anyone help please?

 

[Dick]

If f(z) has a simple pole at z0 then it has the form g(z)/(z-z0) where limit z->z0 g(z)=Res(f,z0). Use that form.

 

[jackmell]

Hi all,
I'm trying to solve 4.15 from the attached file, can anyone help? I tried to use residue thm , i.e the integral of f over the curve gamma-r equals winding number of z0 over gamma-r and residue of z0 of f. I can't see how b-a relates to the winding number of z0. Can anyone help please?

I believe that's not worded properly. I think it should say:

$$\gamma_r:[a,b]\to z_0 +re^{it},\quad a\leq t \leq b$$

and keep in mind if f(z) has a simple pole at z_0, it can be written as:

$$f(z)=\frac{k}{(z-z_0)}+g(z)$$

where g(z) is analytic at z_0 so substitute that expression into the integral, then integrate it directly over the arc between a and b and let the radius r go to zero.

Also, if g(z) is analytic at z_0 then it's bounded say |g(z)|<M so you could just integrate

$$\frac{k}{z-z_0}+M$$

for some finite constant M and then let the radius of the arc go to zero. I mean I'm just about giving it to you right?

 

[hermanni]

Thanx a lot guys , I solved the problem