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Is there a simple way to relate complete elliptic integrals of the first and second kind of the form

$$ K(1/m)\,,\quad K(-m)\,,\quad E(1/m)\,,\quad\text{and}\quad E(-m) $$

to $E(m)$ or $K(m)$. Here, $K(m)$ and $E(m)$ are complete elliptic integrals of the first and second kind, respectively and $m$ is a real number.

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DLMF 19.7.3 gives $$K(1/m)=\sqrt m(K(m)-iK(1-m))$$ $$E(1/m)=\frac1{\sqrt m}(E(m)+iE(1-m)-(1-m)K(m)-imK(1-m))$$ DLMF 19.7.5 gives $$K(-m)=\frac1{\sqrt{1+m}}K\left(\frac m{1+m}\right)$$ $$E(-m)=\sqrt{1+m}E\left(\frac m{1+m}\right)$$ Here $m$ is the parameter.

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