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Math Help - how to determine the complex integral ∫f(e^it) cost dt

  1. #1
    Junior Member sorv1986's Avatar
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    Smile how to determine the complex integral ∫f(e^it) cost dt

    If f(z) is an analytic function, then what will be the value of ∫f(e^it) cost dt,

    ,where (limit of integration) lower limit=0, upper limit =2π.

    Thanks for your help in advance. regards.
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  2. #2
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    Re: how to determine the complex integral ∫f(e^it) cost dt

    Quote Originally Posted by sorv1986 View Post
    If f(z) is an analytic function, then what will be the value of ∫f(e^it) cost dt,

    ,where (limit of integration) lower limit=0, upper limit =2π.

    Thanks for your help in advance. regards.
    \displaystyle \begin{align*} \int_0^{2\pi}{f\left(e^{it}\right)\cos{t}\,dt} &= \int_0^{2\pi}{f\left(e^{it}\right)\left(\frac{e^{i  t} + e^{-it}}{2}\right)dt} \\ &= \int_0^{2\pi}{f\left(e^{it}\right)\left(\frac{1 + e^{-2it}}{2}\right)e^{it}\,dt} \\ &= \frac{1}{2i}\int_0^{2\pi}{f\left( e^{it} \right)\left[1 + \left(e^{it}\right)^{-2} \right] i\,e^{it}\,dt} \end{align*}

    Now make the substitution \displaystyle \begin{align*} u = e^{it} \implies du = i\,e^{it}\,dt \end{align*} and note that \displaystyle \begin{align*}   u(0) = 1\end{align*} and \displaystyle \begin{align*} u(2\pi) = 1 \end{align*} and the integral becomes

    \displaystyle \begin{align*} \frac{1}{2i}\int_1^1{f(u)\left(1 + u^{-2}\right)du} = 0 \end{align*}
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