Equivalence of Definitions of Complete Elliptic Integral of the Second Kind
Jump to navigation
Jump to search
Theorem
The following definitions of the concept of Complete Elliptic Integral of the Second Kind are equivalent:
Definition 1
- $\ds \map E k = \int \limits_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$
is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.
Definition 2
- $\ds \map E k = \int \limits_0^1 \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2} } \rd v$
is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.
Proof
Let $\map E k$ be the complete elliptic integral of the second kind by definition $1$.
Let $v := \sin \phi$.
Then we have:
\(\ds \dfrac {\d v} {\d \phi}\) | \(=\) | \(\ds \cos \phi\) | Derivative of Sine Function | |||||||||||
\(\ds \phi\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \phi\) | \(=\) | \(\ds \dfrac \pi 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds 1\) |
Hence:
\(\ds \map E k\) | \(=\) | \(\ds \int \limits_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \rd \phi\) | Definition 1 of Complete Elliptic Integral of the Second Kind | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \limits_0^1 \sqrt {1 - k^2 \sin^2 \phi} \frac {\d v} {\cos \phi}\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \limits_0^1 \sqrt {1 - k^2 \sin^2 \phi} \frac {\d v} {\sqrt {1 - \sin^2 \phi} }\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \limits_0^1 \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2} } \rd v\) | substituting for $v$ |
Thus $\map E k$ is the complete elliptic integral of the second kind by definition $2$.
$\blacksquare$