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 
and 
around a point 
are...
^{n}}{n!})
(1)
^{n} \frac{(z - z_{0})^{n}}{n!})
(2)
Kind regards
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
