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In the wiki page, complete elliptic integrals of first and second kind are related to the Hypergeometric function via: $$ K(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})\\ E(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},-\frac{1}{2};1;k^{2}) $$

But when evaluate the following command in Mathematica (ver.12):

Pi/2 Hypergeometric2F1[1/2, 1/2, 1, k^2]

I get: EllipticK[k^2]

The same thing happens for $ \mathstrut_{2}F_{1}(\frac{1}{2},-\frac{1}{2};1;k^{2})$ as well. I get EllipticE[k^2] instead of EllipticE[k].

What am I missing ?

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What you are missing is that the Wiki page you refer to uses the modulus convention for elliptic integrals, while Mathematica uses the parameter convention. I had ranted about this at great length here, and see also the docs.

That's why you need to be careful about using any formula you see from other references in general; you need to make sure the convention it uses is the same (or can be converted to) the convention Mathematica uses.

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