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Math Help - Complex Power Series

  1. #1
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    Complex Power Series

    Hi, how are you supposed to represent f(z) = sin(z)/z as a power series of the form sigma c_n z^n ? I have in my notes a formula for this when f is holomorphic in a ball centred at 0, but clearly here f has a singularity at 0 so I don't know what to do.

    Thanks
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  2. #2
    MHF Contributor chisigma's Avatar
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    The complex variable function \sin z can be expressed as 'Weierstrass product' in the following way...

    \sin z = z\cdot \prod_{n=1}^{\infty} (1-\frac{z^{2}}{n^{2} \pi^{2}}) (1)

    That means that \sin z is an entire function, i,e. it is analytic on the whole complex plane. From (1) we obtain...

    \frac{\sin z}{z} = \prod_{n=1}^{\infty} (1-\frac{z^{2}}{n^{2} \pi^{2}}) (2)

    ... so that \frac{\sin z}{z} is also an entire function...

    Kind regards

    \chi \sigma
    Last edited by chisigma; April 20th 2010 at 11:16 AM. Reason: The [orthographic] error pointed out by HallsoIvy has been corrected...
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  3. #3
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    chisigma, Magus01 wants an infinite sum, not an infinite product! Also, by the way, the word in English is "analytic".

    Magus01, the standard power series for sin(z) is \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1}. The power series for \frac{sin(z)}{z} is just that divided, term by term, by z: \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n}.
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