An identity involving complete elliptic integrals of the first and third kind.

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Let $m = 1-k_3^2 = (2+sqrt3)/4$, where $k_3 = (sqrt6 - sqrt2) / 4$ is the third elliptic singular value. So $K(m)/K'(m)=sqrt3$. For this special value of $m$, Mathematica tells me that
$$
K(m) = 4(1-m)Pi(2m-1,m).
$$
Is there any non-numeric reason that this identity should hold for this particular $m$?




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    Let $m = 1-k_3^2 = (2+sqrt3)/4$, where $k_3 = (sqrt6 - sqrt2) / 4$ is the third elliptic singular value. So $K(m)/K'(m)=sqrt3$. For this special value of $m$, Mathematica tells me that
    $$
    K(m) = 4(1-m)Pi(2m-1,m).
    $$
    Is there any non-numeric reason that this identity should hold for this particular $m$?




    elliptic-integrals




    share|cite|improve this question





















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      up vote
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      down vote

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      Let $m = 1-k_3^2 = (2+sqrt3)/4$, where $k_3 = (sqrt6 - sqrt2) / 4$ is the third elliptic singular value. So $K(m)/K'(m)=sqrt3$. For this special value of $m$, Mathematica tells me that
      $$
      K(m) = 4(1-m)Pi(2m-1,m).
      $$
      Is there any non-numeric reason that this identity should hold for this particular $m$?




      elliptic-integrals




      share|cite|improve this question











      Let $m = 1-k_3^2 = (2+sqrt3)/4$, where $k_3 = (sqrt6 - sqrt2) / 4$ is the third elliptic singular value. So $K(m)/K'(m)=sqrt3$. For this special value of $m$, Mathematica tells me that
      $$
      K(m) = 4(1-m)Pi(2m-1,m).
      $$
      Is there any non-numeric reason that this identity should hold for this particular $m$?





      elliptic-integrals





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      asked Jul 17 at 13:31









      Hao Chen

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