2.5 Q20

[Nikki Mai]

Can anyone help me solve 2.5 question20?
I read the question a few times.
i do not know how to do it.
Thank you.

[Aleena Au]

This is my attempt at the question.


Assume f's Laurent series is not unique.

Then, we have
$$f(z) = \sum a_{n} (z-z_{0})^n$$
$$f(z) = \sum b_{n} (z-z_{0})^n$$

Subtract the two equations and get
$$0 = \sum (a_{n}-b_{n}) (z-z_{0})^n$$

So, we must have
$$a_{n} = b_{n}$$ and f's Laurent series must be unique.

[Victor Ivrii]

Aleena It is not the answer because you rely upon unsustained claim that if $f(z)=0$ then it's Laurent's coefficients are $0$ which is an equivalent form of the same question.
Hint: Consider $\int_\gamma (z-z_0)^m  f(z)\,dz $ where $\gamma$ is a counter-clockwise circle around $z_0$ and prove that it is equal to $2\pi i a_{m-1}$.