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Math Help - Complex Taylor series for sin(z)?

  1. #1
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    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 ??
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Complex Taylor series for sin(z)?

    Quote Originally Posted by cassius View Post
    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 e^{i z} and e^{-i z} around a point z_{0} are...

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

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

    Kind regards

    \chi \sigma
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  3. #3
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    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
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