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Math Help - Interesting Integral

  1. #1
    Behold, the power of SARDINES!
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    Interesting Integral

    Show that for a \ne 0, a \in \mathbb{R} that

    \int_{0}^{\pi}\cos(a\sin(\theta))e^{a\cos(\theta)}  d\theta=\pi
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  2. #2
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    Quote Originally Posted by TheEmptySet View Post

    Show that for a \ne 0, a \in \mathbb{R} that \int_{0}^{\pi}\cos(a\sin(\theta))e^{a\cos(\theta)}  d\theta=\pi
    it's nice but probably not hard enough!

    Spoiler:


    e^{ae^{i \theta}}=e^{a \cos \theta}e^{ia\sin \theta}=e^{a \cos \theta}(\cos(a \sin \theta) + i \sin(a \sin \theta)). therefore: e^{a \cos \theta} \cos(a \sin \theta) = \text{Re}(e^{ae^{i \theta}})=\text{Re} \left(\sum_{n=0}^{\infty}\frac{a^ne^{in \theta}}{n!} \right)=\sum_{n=0}^{\infty} \frac{a^n \cos(n \theta)}{n!}.

    hence \int e^{a \cos \theta} \cos(a \sin \theta) \ d \theta = \theta + \sum_{n=1}^{\infty} \frac{a^n \sin(n \theta)}{n \cdot n!} +C. your definite integral is just a simple result of what we have now!
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  3. #3
    Behold, the power of SARDINES!
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    Wow that is a very nice solution. Better than this, but here is another way....

    Consider the integral where \gamma is the unit circle

    \int_\gamma \frac{e^{az}}{z}dz in the complex plane. Then by cauchy's integral formula or the residue theorem.

    \int_\gamma \frac{e^{az}}{z}dz=2\pi i

    Parameterize the curve with z=e^{i\theta} for -\pi \le \theta \le \pi and then dz=ie^{i\theta}d\theta=izd\theta=

    \int_{-\pi}^{\pi} \frac{e^{ae^{i\theta}}}{z}izd\theta=i\int_{-\pi}^{\pi}e^{a\cos(\theta) +ia\sin(\theta)}d\theta=i\int_{-\pi}^{\pi}e^{a\cos(\theta)}[\cos(a\sin(\theta)+i\sin(a\sin(\theta))]d\theta=

    i\int_{-\pi}^{\pi}\cos(a\sin(\theta)e^{a\cos(\theta)}d\the  ta-\int_{-\pi}^{\pi}\sin(a\sin(\theta))e^{a\cos(\theta)}d\th  eta

    So we know that this equals 2\pi i setting immaginary parts equal we get

    2\pi =\int_{-\pi}^{\pi}\cos(a\sin(\theta)e^{a\cos(\theta)}d\the  ta

    Since this is an even function we get (Finally)

    \pi =\int_{0}^{\pi}\cos(a\sin(\theta)e^{a\cos(\theta)}  d\theta
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