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Thread: 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 $\displaystyle \varphi:\mathbb{N}\to\mathbb{R}$ be a function and $\displaystyle n\in\mathbb{N}$.
    For $\displaystyle z\in\mathbb{C}$ define a function $\displaystyle \Phi:\mathbb{C}\to\mathbb{C}$ by
    $\displaystyle \Phi(z):=\prod_{k=1}^{n}\big(1+\varphi(k)z\big)$.
    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 $\displaystyle 1+\varphi(k)z$ as $\displaystyle r_ke^{i\alpha_k}=r_k(\cos\alpha_k+i\sin\alpha_k)$, then the product is $\displaystyle (\prod_{k=1}^nr_k)(\cos(\sum_{k=1}^n\alpha_k)+i\si n(\sum_{k=1}^n\alpha_k))$.

    Now, if $\displaystyle z=re^{i\alpha}$, then $\displaystyle \varphi(k)z=\varphi(k)re^{i\alpha}$. Also, using some trigonometry, one can express $\displaystyle 1+\varphi(k)z$ as $\displaystyle r_ke^{i\alpha_k}$ where $\displaystyle r_k$ and $\displaystyle \alpha_k$ are expressions containing $\displaystyle r$, $\displaystyle \alpha$ and $\displaystyle \varphi(k)$.
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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; Oct 6th 2010 at 06:28 AM.
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