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  1. MHB Apprentice

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    #1
    How to prove:

    $\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $

    where \[ k(\theta)=\sqrt\frac{4Rh \tan \theta}{Z^2+(R+h \tan \theta)^2}\]

    and $ K[k(\theta)] $ is the complete elliptic integral of the first kind, defined by

    \[ K[k(\theta)]= \int_0^{\frac{\pi }{2}}\frac{\,d\phi}{\sqrt{1-k^2(\theta)\sin^2 \phi}}\]

    and h, R and Z $ \gt 0 $

  2. Pessimist Singularitarian
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    #2
    Can you show us what you have tried and where you are stuck? This will give our helpers a better idea how to provide help without perhaps offering suggestions that you may already be trying.

  3. MHB Apprentice

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    #3 Thread Author
    Quote Originally Posted by bshoor View Post
    How to prove:

    $\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $

    where \[ k(\theta)=\sqrt\frac{4Rh \tan \theta}{Z^2+(R+h \tan \theta)^2}\]

    and $ K[k(\theta)] $ is the complete elliptic integral of the first kind, defined by

    \[ K[k(\theta)]= \int_0^{\frac{\pi }{2}}\frac{\,d\phi}{\sqrt{1-k^2(\theta)\sin^2 \phi}}\]

    and h, R and Z $ \gt 0 $
    Please make correction of the post:
    $\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}d\theta=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $

  4. MHB Apprentice

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    #4 Thread Author
    Quote Originally Posted by MarkFL View Post
    Can you show us what you have tried and where you are stuck? This will give our helpers a better idea how to provide help without perhaps offering suggestions that you may already be trying.
    I have tried in different ways. But the integral becomes more and more complicated. Anyway the identity can be modified to :

    $\int_{0}^{\frac{\pi }{2}}{\sqrt {\sin \theta cos\theta}k(\theta) K[k(\theta)]}d\theta=\pi \sqrt{ \frac{Rh}{R^2 + (h+Z)^2}} $

    by multiplying both side by $ 2 \sqrt{Rh} $

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