[Complex analysis] Coefficients of Laurent series

[mick25]

Homework Statement

I have some past exam questions that I am confused with

Homework Equations

a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz

The Attempt at a Solution

I'm not sure how to approach this, I'm completely lost and just attempted to solve a few:

a) it says f(z) has a pole of order 5, so f(z) = \frac{g(z)}{z^5}, g(z)\neq0

so then I guess the condition is a_{4} = \frac{g^{(4)}(0)}{4!}? But that's just applying the formula for the coefficients...

c) f(\frac{1}{z}) = \frac{g(\frac{1}{z})}{z^5} => f(z) = z^5g(z)

so the coefficients are a_{n} = \frac{1}{2\pi i} \oint_\gamma z^5g(z) dz?

d) \frac{1}{f(z)} = \frac{g(z)}{z^5} => f(z) = \frac{z^5}{g(z)}

so, a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{z^5}{g(z)} dz

g) a_{-1} = \frac{1}{2\pi i} \oint_ \gamma f(z) dz = \frac{1}{2\pi i} = Res(f; c)*I(\gamma; c) = -Res(f; c)

h) \frac{a_{n}}{16} = 4^{n}a_{n} => 0 = a_{n}(4^{n} - 4^{-2}) => a_{n} = 0 or n = -2for e) and f), I'm not sure what the relevance of the essential singularity is

Well, I think you can see I'm clearly lost, would appreciate if you could help me out.

 

[jackmell]

Homework Statement

I have some past exam questions that I am confused with

Homework Equations

a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz

The Attempt at a Solution

I'm not sure how to approach this, I'm completely lost and just attempted to solve a few:

a) it says f(z) has a pole of order 5, so f(z) = \frac{g(z)}{z^5}, g(z)\neq0

so then I guess the condition is a_{4} = \frac{g^{(4)}(0)}{4!}? But that's just applying the formula for the coefficients...

c) f(\frac{1}{z}) = \frac{g(\frac{1}{z})}{z^5} => f(z) = z^5g(z)

so the coefficients are a_{n} = \frac{1}{2\pi i} \oint_\gamma z^5g(z) dz?

d) \frac{1}{f(z)} = \frac{g(z)}{z^5} => f(z) = \frac{z^5}{g(z)}

so, a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{z^5}{g(z)} dz

g) a_{-1} = \frac{1}{2\pi i} \oint_ \gamma f(z) dz = \frac{1}{2\pi i} = Res(f; c)*I(\gamma; c) = -Res(f; c)

h) \frac{a_{n}}{16} = 4^{n}a_{n} => 0 = a_{n}(4^{n} - 4^{-2}) => a_{n} = 0 or n = -2




for e) and f), I'm not sure what the relevance of the essential singularity is

Well, I think you can see I'm clearly lost, would appreciate if you could help me out.

You need to look at these by sayin' for example, (a), if
f(z)=\sum_{n=0}^{\infty} a_n z^n+\sum_{n=1}^{\infty}\frac{b_n}{z^n}
and it has a pole of order 5, then that must mean it at least has a non-zero term for the \frac{1}{z^5} term and all the other terms \frac{1}{z^n} for n>5 must be zero else the pole would have a higher order. Ok, I'll do some of (b) same dif:

If:
f(z)=\sum_{n=0}^{\infty} a_n z^n+\sum_{n=1}^{\infty}\frac{b_n}{z^n}
and f(z)-7e^{1/z} has a pole of order 5 and I know that e^{1/z}=\sum_{n=0}^{\infty} \frac{1}{z^n n!}, then that must mean the terms of f(z) would have to be what to cancel all that out except for at least the \frac{1}{z^5} term?

Ok then, do that same thing with all the rest.