19 Elliptic IntegralsLegendre’s Integrals19.2 Definitions19.4 Derivatives and Differential Equations

§19.3 Graphics

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Keywords:

Legendre’s elliptic integrals, graphics

Permalink:

http://dlmf.nist.gov/19.3

See also:

Annotations for Ch.19

Contents

1. §19.3(i) Real Variables
2. §19.3(ii) Complex Variables

§19.3(i) Real Variables

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Notes:

These graphics were produced at NIST.

Permalink:

http://dlmf.nist.gov/19.3.i

See also:

Annotations for §19.3 and Ch.19

See Figures 19.3.1–19.3.6 for complete and incomplete Legendre’s elliptic integrals.

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind and $k$: real or complex modulus

Referenced by:

§19.3(i)

Permalink:

http://dlmf.nist.gov/19.3.F1

Encodings:

pdf, png

See also:

Annotations for §19.3(i), §19.3 and Ch.19

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln z$: principal branch of logarithm function

Referenced by:

§19.17

Permalink:

http://dlmf.nist.gov/19.3.F2

Encodings:

pdf, png

See also:

Annotations for §19.3(i), §19.3 and Ch.19

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.3.F3

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(i), §19.3 and Ch.19

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.3.F4

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(i), §19.3 and Ch.19

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.3.F5

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(i), §19.3 and Ch.19

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\csc z$: cosecant function, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.3(i)

Permalink:

http://dlmf.nist.gov/19.3.F6

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(i), §19.3 and Ch.19

§19.3(ii) Complex Variables

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Keywords:

Legendre’s elliptic integrals, graphics

Notes:

These graphics were produced at NIST.

Permalink:

http://dlmf.nist.gov/19.3.ii

See also:

Annotations for §19.3 and Ch.19

In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part and $k$: real or complex modulus

Referenced by:

§19.17, §19.17, §19.3(ii)

Permalink:

http://dlmf.nist.gov/19.3.F7

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(ii), §19.3 and Ch.19

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part and $k$: real or complex modulus

Referenced by:

§19.3(ii)

Permalink:

http://dlmf.nist.gov/19.3.F8

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(ii), §19.3 and Ch.19

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.3.F9

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(ii), §19.3 and Ch.19

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.3.F10

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(ii), §19.3 and Ch.19

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.3.F11

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(ii), §19.3 and Ch.19

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.17, §19.17

Permalink:

http://dlmf.nist.gov/19.3.F12

Encodings:

Vizualization, pdf, png

See also:

Annotations for §19.3(ii), §19.3 and Ch.19

19.2 Definitions19.4 Derivatives and Differential Equations

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