Elliptic Integral Equation Proof

bb8

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Jan 19, 2016
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How can I start to prove the equation:
K'(k) = E(k)/k(1 − k^2)−K(k)/k for all 0 < k < 1
by using the complete elliptic integrals of the first and second kind?
I have already changed the integral from (0 to pi/2) to (0 to 1) for both K(k) and E(k), but I'm not sure where to go from there.
Thank you!
 
How can I start to prove the equation:
K'(k) = E(k)/k(1 − k^2)−K(k)/k for all 0 < k < 1
by using the complete elliptic integrals of the first and second kind?
I have already changed the integral from (0 to pi/2) to (0 to 1) for both K(k) and E(k), but I'm not sure where to go from there.
Thank you!


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I have changed
K(k)=(integral 0 to pi/2) 1/sqrt(1-k^2sin^2x) dx to K(k)=(integral 0 to 1) 1/[2cosx(sqrt(t-k^2t^2))] dt
and
E(k)=(integral 0 to pi/2) sqrt(1-k^2sin^2x) dx to E(k)=(integral 0 to 1) [sqrt(1-k^2t)]/[2(sqrt(t))cosx] dt
by performing the change t=sin^2(x), dt=2sinxcosx dx and sqrt(t)=sinx
 
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