1. ## Laurent series expansion

Give the Laurent series expansion for the function $f(z)$ centered at $z_0$. State the possible domains.

$f(z) = e^z\sin z$ $z_0=0$

Here, I see I will get two series multiplied together. I do not see how to get one Laurent series expansion for this one though. Any hints would be nice. Thanks in advance.

2. Originally Posted by pascal4542
Give the Laurent series expansion for the function $f(z)$ centered at $z_0$. State the possible domains.

$f(z) = e^z\sin z$ $z_0=0$

Here, I see I will get two series multiplied together. I do not see how to get one Laurent series expansion for this one though. Any hints would be nice. Thanks in advance.
What you will get is a Maclaurin series.

Perhaps the question meant $f(z) = \frac{e^z}{\sin z}$ $z_0=0$ ....

3. Originally Posted by mr fantastic
What you will get is a Maclaurin series.

Perhaps the question meant $f(z) = \frac{e^z}{\sin z}$ $z_0=0$ ....

Maybe the question I have was what I typed before. I will have to ask for sure though. So, I will assume for now that it is the fraction so that I will get the Laurent series it is asking for.

4. I was wondering if someone can check this.

From what I understand this is the Taylor series expansion. However, the problem is asking for the Laurent series and the domain. So far I have

$e^z \sin(z) =$

$(1+z+z^2/2+z^3/6+z^4/24+\cdots)(z-z^3/6+z^5/120-z^7/5040+\cdots)$
$
=z+z^2+z^5/120-z^7/5040+\cdots.$
However, this is not in series notation and I do not know the domain either. I am stuck here. I need some direction on how to get the Laurent series. Thank you.