Amplitude: Series representations (formula 09.24.06.0008)

JacobiAmplitude

$Mathematica Notation$

Elliptic Functions

JacobiAmplitude[z,m]

Series representations

Generalized power series

Expansions at z==0

http://functions.wolfram.com/09.24.06.0008.01

Input Form

JacobiAmplitude[z, m] \[Proportional] z - (m z^3)/6 + (1/120) (4 m + m^2) z^5 - ((16 m + 44 m^2 + m^3) z^7)/5040 + ((64 m + 912 m^2 + 408 m^3 + m^4) z^9)/362880 - ((256 m + 15808 m^2 + 30768 m^3 + 3688 m^4 + m^5) z^11)/39916800 + ((1024 m + 259328 m^2 + 1538560 m^3 + 870640 m^4 + 33212 m^5 + m^6) z^13)/ 6227020800 - (1/1307674368000) ((4096 m + 4180992 m^2 + 65008896 m^3 + 106923008 m^4 + 22945056 m^5 + 298932 m^6 + m^7) z^15) + (1/355687428096000) ((m^8 + 2690416 m^7 + 586629984 m^6 + 6337665152 m^5 + 9860488448 m^4 + 2536974336 m^3 + 67047424 m^2 + 16384 m) z^17) - ((m^9 + 24213776 m^8 + 14804306080 m^7 + 345558617984 m^6 + 1165333452544 m^5 + 782931974144 m^4 + 95153582080 m^3 + 1073463296 m^2 + 65536 m) z^19)/121645100408832000 + O[z^11]

Standard Form

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MathML Form

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</mo> <msup> <mi> m </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6337665152 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9860488448 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2536974336 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 67047424 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16384 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 17 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 121645100408832000 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> m </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 24213776 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 14804306080 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 345558617984 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1165333452544 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 782931974144 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 95153582080 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1073463296 </mn> <mo> ⁢ </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 65536 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 19 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 11 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> JacobiAmplitude </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 120 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> m </ci> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 44 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> <apply> <power /> <cn type='integer'> 5040 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 64 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 912 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 408 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> <apply> <power /> <cn type='integer'> 362880 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 256 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 15808 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 30768 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3688 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 11 </cn> </apply> <apply> <power /> <cn type='integer'> 39916800 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1024 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 259328 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1538560 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 870640 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> 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<power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 298932 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 15 </cn> </apply> <apply> <power /> <cn type='integer'> 1307674368000 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 355687428096000 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> m </ci> <cn type='integer'> 8 </cn> </apply> <apply> <times /> <cn type='integer'> 2690416 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 586629984 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6337665152 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9860488448 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2536974336 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 67047424 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 16384 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 17 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 121645100408832000 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 24213776 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 14804306080 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 345558617984 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1165333452544 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 782931974144 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 95153582080 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1073463296 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 65536 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 19 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 11 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02