Modular lambda function: Series representations (formula 09.51.06.0006)

ModularLambda

$Mathematica Notation$

Elliptic Functions

ModularLambda[z]

Series representations

Generalized power series

Expansions at z==infinity

http://functions.wolfram.com/09.51.06.0006.01

Input Form

ModularLambda[z] \[Proportional] 16 E^(I Pi z) - 128 E^(2 I Pi z) + 704 E^(3 I Pi z) - 3072 E^(4 I Pi z) + 11488 E^(5 I Pi z) - 38400 E^(6 I Pi z) + 117632 E^(7 I Pi z) - 335872 E^(8 I Pi z) + 904784 E^(9 I Pi z) - 2320128 E^(10 I Pi z) + 5702208 E^(11 I Pi z) - 13504512 E^(12 I Pi z) + 30952544 E^(13 I Pi z) - 68901888 E^(14 I Pi z) + 149403264 E^(15 I Pi z) - 316342272 E^(16 I Pi z) + 655445792 E^(17 I Pi z) - 1331327616 E^(18 I Pi z) + 2655115712 E^(19 I Pi z) - 5206288384 E^(20 I Pi z) + O[E^(21 I Pi z)] /; Im[z] > 0 && (Abs[z] -> Infinity)

Standard Form

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MathML Form

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</mrow> <mo> + </mo> <mrow> <mn> 2655115712 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 19 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5206288384 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 20 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 128 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 704 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3072 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 11488 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 5 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 38400 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 6 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 117632 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 7 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 335872 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 8 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 904784 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 9 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2320128 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 10 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5702208 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 11 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 13504512 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 12 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 30952544 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 13 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 68901888 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 14 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 149403264 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 15 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 316342272 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 16 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 655445792 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 17 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1331327616 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 18 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2655115712 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 19 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5206288384 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 20 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 21 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02