MHB Finding Taylor Series of $\dfrac{1}{z-i} \div \left(z+i\right)$

[Dustinsfl]

I am trying to find the Taylor series for
$$\displaystyle
\dfrac{\left(\dfrac{1}{z-i}\right)}{z+i}
$$
where z is a complex number.There is a reason it is set up as a fraction over the denominator so let's not move it down.

 

[HallsofIvy]

Well, the first thing I would do is go ahead and do this division- $$\frac{\frac{1}{z- i}}{z+ i}= \frac{1}{z^2+ 1}$$

Then I would rewrite it as $$\frac{1}{1-(-z^2)}$$ and use the formula for the sum of a geometric series:
$$\sum r^n= \frac{1}{1- r}$$

 

[Dustinsfl]

Well, the first thing I would do is go ahead and do this division- $$\frac{\frac{1}{z- i}}{z+ i}= \frac{1}{z^2+ 1}$$

Then I would rewrite it as $$\frac{1}{1-(-z^2)}$$ and use the formula for the sum of a geometric series:
$$\sum r^n= \frac{1}{1- r}$$

Thanks, I was trying to do something a little different with it but that will suit the objective.

 

[HallsofIvy]

It was after posting that I noticed your "There is a reason it is set up as a fraction over the denominator so let's not move it down". That seems strange. What was your reason? Of course, the Taylor's series (about 0) will be the same however you write the function.

 

[Dustinsfl]

It was after posting that I noticed your "There is a reason it is set up as a fraction over the denominator so let's not move it down". That seems strange. What was your reason? Of course, the Taylor's series (about 0) will be the same however you write the function.

It had to do with a contour integral and I was going to integrate it as two separate integrals, because I thought it would work out nicer that way. However, doing it in this manner is fine. I was trying to fit a different form but this was obviously easier and still fit the right form in the end.