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ID: INPUT{id.yaml}

Title: >
  Complete elliptic integral of the third kind $\Pi(n,m)$

Definition: >
  The list contains the evaluations of the
  complete elliptic integral of the third kind
  $\Pi(n,m) = \int_0^{\pi/2} \frac{1}{(1-n\sin^2 \theta)\sqrt{1-m\sin^2 \theta}} d\theta$
  for some parameters $n$ and $m=k^2$,
  where $k$ is the ellpitic modulus.

Parameters:
  n:
    display: $n$
    type: R
    constraints: $0\leq n < 1$
  m:
    display: $m$
    type: R
    constraints: $0\leq m < 1$
   
Comments:
  comment-k: >
    $\Pi(n,m)$ is also denoted as $\Pi(n,k)$ despite the relation 
    $m=k^2$ between the parameter $m$ and the elliptic modulus $k$.
  comment-sign-of-n: >
    $\Pi(n,m)$ is sometimes defined with the inverse sign of 
    the characteristic $n$.
  comment-sign-of-n: >
    The parameter $n$ can be larger than $1$, 
    although in that case one has to specify how to integrate
    through the singulatiry of the integrand, 
    e.g. by taking the Cauchy principal value.
  
Formulas:

Programs:

References:

Links:
  Wiki:
    title: "Wikipedia: Complete elliptic integral of the third kind"
    url: https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_third_kind
    
Similar tables:
  
Keywords:
  
Tags:
- period
- elliptic curves

Data properties:
  type: R
  complete: no

Display properties:
  number-header: $\Pi(n,m)$

Numbers: INPUT{numbers.yaml}

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Complete elliptic integral of the third kind $\Pi(n,m)$

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Numbers

$n$

$m$

$\Pi(n,m)$

INPUT{numbers.yaml} (not shown in preview)

Definition

The list contains the evaluations of the complete elliptic integral of the third kind $\Pi(n,m) = \int_0^{\pi/2} \frac{1}{(1-n\sin^2 \theta)\sqrt{1-m\sin^2 \theta}} d\theta$ for some parameters $n$ and $m=k^2$, where $k$ is the ellpitic modulus.

Parameters

$n$

— real number ($0\leq n < 1$)

$m$

— real number ($0\leq m < 1$)

Comments

(1)

$\Pi(n,m)$ is also denoted as $\Pi(n,k)$ despite the relation $m=k^2$ between the parameter $m$ and the elliptic modulus $k$.

(2)

The parameter $n$ can be larger than $1$, although in that case one has to specify how to integrate through the singulatiry of the integrand, e.g. by taking the Cauchy principal value.

Links

[1]

Wikipedia: Complete elliptic integral of the third kind

Data properties

Entries are of type: real number

Table is complete: no