Laurent Series of $f(z)=(z+1)\sin\frac{z^{2}+2z+5}{(z+1)^{2}}$

Question

We are asked to find the Laurent series for the following function.

$$f(z)=(z+1)\sin\frac{z^{2}+2z+5}{(z+1)^{2}}$$

Around the point $$z_{0}=-1$$

I have tried to factor the inside of sine, to no avail.

I know that $\sin(z)$ at $z=0$ is a simple pole, even if the inside is a square like $(z+1)^2$.

Furthermore, I think term $(z+1)$ multiplied by sine, might be suggestive of a derivative of some sort.

Other than that, I am lost as far as what to do with the inside of sine.

Any help would be appreciated!

Answer 1

Here are some hints:

1. $z^2+2z+5 = (z+1)^2+4$
2. Use a Taylor series of sine, but not the usual one.

I think this should be enough to get you started. If you have no idea how to use these hints, I can give you more hints.

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Edit #1.

I thought making "one" italics would be a good hint...

Basically, I was suggesting that you use the Taylor series of sin centered at 1.

But @Santosh Linkha's comment might work better, since you'll be able to use the Maclaurin series of sin and cos.

In either case, you'll have $\sin(1)$ and $\cos(1)$ appear as coefficients of your series.