For the complete elliptic integral of the first kind, we find on Wikipedia:
$$
K(k) = \int_\limits{0}^{\pi/2} \hspace{-1ex} \frac{d\theta}
{\sqrt{1 - k^2 \sin^2 \theta}}
$$
But in Abramowitz and Stegun 17.3.1 we find:
$$
K(m) = \int_\limits{0}^{\pi/2} \hspace{-1ex} \frac{d\theta}
{\sqrt{1 - m \sin^2 \theta}}
$$
which is not the same.
The definition used by Mathematica for EllipticK is that of Abramowitz and Stegun.
Why does Wikipedia have a different definition?
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1$\begingroup$ Definitions are correct by definition. There are no “correct” definitions in an absolute sense. $\endgroup$– David HCommented May 4 at 16:58
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$\begingroup$ Unless of course they are ambiguous. Or inconsistent with already existing definitions. Or circular (without reason). Or given in terms of undefined concepts. Or maybe some other things, but apart from all that you are right! youtu.be/Qc7HmhrgTuQ?t=13 $\endgroup$– Jos BergervoetCommented May 4 at 17:42
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2 Answers
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The standard definition used by Legrende, Cayley and others is the one from Wikipedia. Table of Integrals, Series, and Products by Gradshteyn and Ryzhik in the seventh edition also uses that notation. Just keep in mind that $K_{Mathematica}(v^2) = K_{Wikipedia}(v) $.
Here and here Wolfram uses the standard definition but added the following warning for the incomplete elliptic integral:
"The elliptic integral of the first kind is implemented in the Wolfram Language as EllipticF[phi, m] (note the use of the parameter m=k^2 instead of the modulus k)."
While for the complete added:
"It is implemented in the Wolfram Language as EllipticK[m], where m=k^2 is the parameter"
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They are the same definition, just with different inputs. Each input style has its own advantages: $K(k^2)$ facilitates Landen and Gauss transformations (which were important for calculating elliptic integrals by hand before digital computers came along), while $K(m)$ is of course easier to type.
answered Apr 18 at 4:42
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$\begingroup$ They aren't the same definition, just like $f(x)=\sqrt{x}$ is not the same function as $f(x)=x$. And you can also not argue that the arguments are given in different units, like for $\sin(x)$ where we can have degrees or radians. And even in that latter case most people would probably agree that there is one correct "official" definition: using radians (so if you use degrees you must clearly notify the readers!) So I'm still wondering, which definition of the elliptic integral can be silently used and in which case do we have to warn the readers? (Or is it so unclear that we always have to?!) $\endgroup$ Commented Apr 18 at 7:39
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