# Toronto Math Forum

## MAT334-2018F => MAT334--Lectures & Home Assignments => Topic started by: Nikki Mai on November 13, 2018, 12:52:41 PM

Title: 2.5 Q20
Post by: Nikki Mai on November 13, 2018, 12:52:41 PM
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.
Title: Re: 2.5 Q20
Post by: Aleena Au on November 13, 2018, 03:09:12 PM
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.
Title: Re: 2.5 Q20
Post by: Victor Ivrii on November 13, 2018, 04:37:12 PM
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 $2\pi i a_{m-1}$.