Dedekind eta modular function: Identities (formula 09.49.17.0010)

DedekindEta

$Mathematica Notation$

Elliptic Functions

DedekindEta[z]

Identities

Functional identities

http://functions.wolfram.com/09.49.17.0010.01

Input Form

DedekindEta[(b + a z)/(d + c z)] == UnitStep[c] Exp[Pi I ((a + d)/(12 c) + s[-d, c])] Sqrt[(-I) (d + c z)] DedekindEta[z] + UnitStep[-c] Exp[Pi I ((a + d)/(12 c) + s[d, -c])] Sqrt[I (d + c z)] DedekindEta[z] /; Element[a, Integers] && Element[b, Integers] && Element[c, Integers] && Element[d, Integers] && a d - b c == 1 && s[h, l] == Sum[(k/l) ((k h)/l - Floor[(k h)/l] - 1/2), {k, 1, l - 1}]

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox[FractionBox[RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mrow> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> d </mi> </mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> s </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> , </mo> <mrow> <mo> - </mo> <mi> c </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <mi> η </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Eta]", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[DedekindEta[Slot[1]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> d </mi> </mrow> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> s </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - 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</mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> s </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> h </mi> <mo> , </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> l </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> l </mi> </mfrac> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> h </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mi> l </mi> </mfrac> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> h </mi> </mrow> <mi> l </mi> </mfrac> <mo> ⌋ </mo> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DedekindEta </ci> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <apply> <exp /> <apply> <times /> <pi /> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> d </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> s </ci> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> DedekindEta </ci> <ci> z </ci> </apply> <apply> <exp /> <apply> <times /> <pi /> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> d </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 12 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <ci> c </ci> </apply> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <ci> c </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> a </ci> <integers /> </apply> <apply> <in /> <ci> b </ci> <integers /> </apply> <apply> <in /> <ci> c </ci> <integers /> </apply> <apply> <in /> <ci> d </ci> <integers /> </apply> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> s </ci> <ci> h </ci> <ci> l </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> l </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> l </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> <apply> <plus /> <apply> <times /> <ci> h </ci> <ci> k </ci> <apply> <power /> <ci> l </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <ci> k </ci> <ci> h </ci> <apply> <power /> <ci> l </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29