Integral from 0 to 1 of Complete Elliptic Integral of First Kind

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Theorem

Let $G$ denote Catalan's constant.

Then:

$2 G = \ds \int_0^1 \map K k \rd k$

where $\map K k$ denotes the complete elliptic integral of the first kind:

$\map K k = \ds \int \limits_0^{\pi / 2} \dfrac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$


Proof

\(\ds \int_0^1 \map K k \rd k\) \(=\) \(\ds \int_0^1 \int_0^{\pi / 2} \dfrac 1 {\sqrt {1 - k^2 \sin^2 \phi} } \rd \phi \rd k\) Definition of Complete Elliptic Integral of the First Kind
\(\ds \) \(=\) \(\ds \int_0^{\pi / 2} \int_0^1 \dfrac 1 {\sqrt {1 - k^2 \sin^2 \phi} } \rd k \rd \phi\) Tonelli's Theorem
\(\ds \) \(=\) \(\ds \int_0^{\pi / 2} \dfrac 1 {\sin \phi} \int_0^{\sin \phi} \dfrac 1 {\sqrt {1 - u^2} } \rd u \rd \phi\) Substitution of $u = k \sin \phi$ and $\rd u = \rd k \sin \phi$
\(\ds \) \(=\) \(\ds \int_0^{\pi / 2} \dfrac {\map \arcsin {\sin \phi} } {\sin \phi} \rd \phi\) Arcsine as Integral
\(\ds \) \(=\) \(\ds \int_0^{\pi / 2} \dfrac \phi {\sin \phi} \rd \phi\) Definition of Inverse Sine
\(\ds \) \(=\) \(\ds 2 \int_0^{\pi / 4} \dfrac {2 \theta} {\sin 2 \theta} \rd \theta\) Substitution of $2 \theta = \phi$ and $2 \rd \theta = \rd \phi$
\(\ds \) \(=\) \(\ds 2 \int_0^{\pi / 4} \dfrac {\theta} {\sin \theta \cos \theta} \rd \theta\) Double Angle Formula for Sine
\(\ds \) \(=\) \(\ds 2 \int_0^{\pi / 4} \dfrac {\theta} {\sin \theta \times \dfrac {\cos \theta} {\cos \theta} \times \cos \theta} \rd \theta\) multiplying by $1$
\(\ds \) \(=\) \(\ds 2 \int_0^{\pi / 4} \dfrac \theta {\tan \theta} \sec^2 \theta \rd \theta\)
\(\ds \) \(=\) \(\ds 2 \int_0^1 \dfrac {\arctan t} t \rd t\) Substitution of $t = \tan \theta$ and $\rd t = \sec^2 \theta \rd \theta$
\(\ds \) \(=\) \(\ds 2 \int_0^1 \dfrac 1 t \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {t^{2 n + 1} } {2 n + 1} \rd t\) Power Series Expansion for Real Arctangent Function
\(\ds \) \(=\) \(\ds 2 \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^n} {2 n + 1} \int_0^1 t^{2 n} \rd t\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds 2 \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^n} {\paren {2 n + 1}^2}\) Integral of Power
\(\ds \) \(=\) \(\ds 2 G\) Definition of Catalan's Constant

$\blacksquare$


Sources