Complex Taylor series for sin(z)?

I know the general formula for a Taylor series about z_0 (z subscript 0) in the complex plane, here http://upload.wikimedia.org/wikipedi...279ed2da10.png, and the Taylor formula for sinx in the real plane, http://upload.wikimedia.org/wikipedi...75a8fd748a.png

i also know the expansion for e^z about z_0 in the complex plane

now i need to work out the Taylor formula formula for sin(z) about the point z_0, not zero. i can see it should resemble the real formula, but i don't know how to approach it.

would you use the definition of complex sine = (e^iz - e^(-iz))/2i and go on from there ??

Re: Complex Taylor series for sin(z)?

Quote:

Originally Posted by

**cassius** I know the general formula for a Taylor series about z_0 (z subscript 0) in the complex plane, here

http://upload.wikimedia.org/wikipedi...279ed2da10.png, and the Taylor formula for sinx in the real plane,

http://upload.wikimedia.org/wikipedi...75a8fd748a.png
i also know the expansion for e^z about z_0 in the complex plane

now i need to work out the Taylor formula formula for sin(z) about the point z_0, not zero. i can see it should resemble the real formula, but i don't know how to approach it.

would you use the definition of complex sine = (e^iz - e^(-iz))/2i and go on from there ??

Your approach is fully correct... take into account that the taylor series expansion of $\displaystyle e^{i z}$ and $\displaystyle e^{-i z}$ around a point $\displaystyle z_{0}$ are...

$\displaystyle e^{i z}= e^{i z_{0}}\ \sum_{n=0}^{\infty} i^{n} \frac{(z - z_{0})^{n}}{n!}$ (1)

$\displaystyle e^{-i z}= e^{-i z_{0}}\ \sum_{n=0}^{\infty} (-i)^{n} \frac{(z - z_{0})^{n}}{n!}$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

Re: Complex Taylor series for sin(z)?

Dear Chisigma,

you kindly answered my question about the taylor expansion of complex sine, http://www.mathhelpforum.com/math-he...-z-192734.html

i made some progress on the problem, but the result i got is a rather long expression, i attach the image here http://savepic.su/813764.gif

is there any way it can be simplified?

thank you so much for your help