In Wolfram's website specific values of incomplete elliptic integral of first, second and third kind are given in terms of complete elliptic integrals of first, second and third kind.

For the incomplete integral of first kind $\text{F}(z, m)$, the site lists the following relations:

$$ \text{F}(\pi/2, m)= \text{K}(m)\\ \text{F}(\text{csc}^{-1}(\sqrt{m}), m)= \frac{1}{\sqrt{m}}\text{K}\left(\frac{1}{m}\right) $$

Similarly for incomplete elliptic integral of second kind $\text{E}(z,m)$:

$$ \text{E}(\pi/2, m)= \text{E}(m)\\ \text{E}(\text{csc}^{-1}(\sqrt{m}), m)= \sqrt{m} \left( \text{E}\left(\frac{1}{m}\right)+ \left(\frac{1}{m}-1\right)\text{K}\left(\frac{1}{m}\right)\right) $$

and for the third kind $\Pi(n,\, z,\,m)$:

$$ \Pi(n,\,\pi/2,\, m)=\Pi(n,\,m)\\ \Pi(n,\, \text{csc}^{-1}(\sqrt{m}),\,m)=\frac{1}{\sqrt{m}}\Pi\left(\frac{n}{m},\,\frac{1}{m}\right) $$

I would like to use this identities in a calculation however the site does not list references. I would appreciate a comprehensive reference list for these relations. Thanks in advance.


1 Answer 1


A good list of relations involving elliptic integrals is to be found in the DLMF. Your examples are obtained by applying the reciprocal modulus/parameter transformation with the amplitude $\phi=\frac\pi2$, which makes the integral(s) on one side complete.


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