### Author Topic: 2.5 Q20  (Read 1351 times)

#### Nikki Mai

• Jr. Member
•  • Posts: 12
• Karma: 1 ##### 2.5 Q20
« 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.

#### Aleena Au

• Newbie
• • Posts: 1
• Karma: 0 ##### Re: 2.5 Q20
« Reply #1 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.

#### Victor Ivrii ##### Re: 2.5 Q20
« Reply #2 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 $2\pi i a_{m-1}$.