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ID: INPUT{id.yaml}

Title: >
  Complete elliptic integral of the second kind $E(m)$

Definition: >
  The list contains the evaluations of the
  complete elliptic integral of the second kind
  $E(m) = \int_0^1 \frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} dt$
  for some parameters $m=k^2$,
  where $k$ is the ellpitic modulus.

Parameters:
  m:
    display: $m$
    type: R
    constraints: $0\leq m \leq 1$
   
Comments:
  comment-k: >
    $E(m)$ is also denoted as $E(k)$ despite the relation 
    $m=k^2$ between the parameter $m$ and the elliptic modulus $k$.
  
Formulas: 
  formula-sin: >
    $E(m) = \int_0^{\pi/2} \sqrt{1-m\sin^2\theta} d\theta$.

Programs:

References:

Links:
  Wiki:
    title: "Wikipedia: Complete elliptic integral of the second kind"
    url: https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind

Similar tables:
  
Keywords:
  
Tags:
- period
- elliptic curves

Data properties:
  type: R
  complete: no

Display properties:
  number-header: $E(m)$

Numbers: INPUT{numbers.yaml}

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Complete elliptic integral of the second kind $E(m)$

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Numbers

$m$

$E(m)$

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

Definition

The list contains the evaluations of the complete elliptic integral of the second kind $E(m) = \int_0^1 \frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} dt$ for some parameters $m=k^2$, where $k$ is the ellpitic modulus.

Parameters

$m$

— real number ($0\leq m \leq 1$)

Formulas

(1)

$E(m) = \int_0^{\pi/2} \sqrt{1-m\sin^2\theta} d\theta$.

Comments

(2)

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

Links

[1]

Wikipedia: Complete elliptic integral of the second kind

Data properties

Entries are of type: real number

Table is complete: no