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I have run into issues in which I have found an incomplete elliptic integral of the first kind represented in multiple ways. In one instance there is a $k^2$ in the denominator and in the other, it is simply a $k$. So my question then is which of these two representations is correct and why there are differences?

$\int_0^\psi \frac{1}{\sqrt{1-k^2sin^2(\theta)}}$ vs. $\int_0^\psi \frac{1}{\sqrt{1-ksin^2(\theta)}}$

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Unfortunately, there are different conventions for the elliptic integrals. Gradshteyn and Ryzhik use one (which is followed by Maple), Abramowitz and Stegun use the other (which is followed by Mathematica). So Mathematica's $E(m)$ is Maple's $EllipticE(\sqrt{m})$.

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The usual definition is with $k^2$: https://dlmf.nist.gov/19.2#ii.

But writing $k$ or $\lambda$ or whatever instead of $k^2$ is no sin, provided the rest of the usage is coherent (in particular, the constant must be constrained to be positive).

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