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Math Help - Real and Imaginary Parts of a Complex Product

  1. #1
    Senior Member bkarpuz's Avatar
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    Real and Imaginary Parts of a Complex Product

    Dear MHF users,

    let \varphi:\mathbb{N}\to\mathbb{R} be a function and n\in\mathbb{N}.
    For z\in\mathbb{C} define a function \Phi:\mathbb{C}\to\mathbb{C} by
    \Phi(z):=\prod_{k=1}^{n}\big(1+\varphi(k)z\big).
    How can I decompose this type of functions into its real and imaginary parts?
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  2. #2
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    If you could represent 1+\varphi(k)z as r_ke^{i\alpha_k}=r_k(\cos\alpha_k+i\sin\alpha_k), then the product is (\prod_{k=1}^nr_k)(\cos(\sum_{k=1}^n\alpha_k)+i\si  n(\sum_{k=1}^n\alpha_k)).

    Now, if z=re^{i\alpha}, then \varphi(k)z=\varphi(k)re^{i\alpha}. Also, using some trigonometry, one can express 1+\varphi(k)z as r_ke^{i\alpha_k} where r_k and \alpha_k are expressions containing r, \alpha and \varphi(k).
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  3. #3
    Senior Member bkarpuz's Avatar
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    Thanks for the information but this seems to be useless for me. :s

    But but but let me think it may give a hand…
    Last edited by bkarpuz; October 6th 2010 at 06:28 AM.
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