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http://functions.wolfram.com/10.06.03.0059.01
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LerchPhi[z, 1, a] == (Hypergeometric2F1[1, a, 1 + a, z]
(1 - 2 UnitStep[-Re[a]]))/a +
(2 ((z Hypergeometric2F1[1, 1 + a - Ceiling[Re[a]], 2 + a - Ceiling[Re[a]],
z])/(1 + a - Ceiling[Re[a]]) + ((1 - Ceiling[Re[a]] + Floor[Re[a]])
UnitStep[Im[a]])/Sqrt[(a - Ceiling[Re[a]])^2]) UnitStep[-Re[a]])/
z^Ceiling[Re[a]]
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Cell[BoxData[RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "1", ",", "a"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "a", ",", RowBox[List["1", "+", "a"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]], ")"]]]], "a"], "+", RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["z", " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", RowBox[List["2", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", "z"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "a", "]"]], "]"]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["1", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo>  </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌉ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌉ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌉ </mo> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "2"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌉ </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </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> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌉ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mfrac> </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> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌉ </mo> </mrow> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mi> a </mi> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> a </mi> </mrow> <mo> ; </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "1"]], Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LerchPhi </ci> <ci> z </ci> <cn type='integer'> 1 </cn> <ci> a </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <apply> <floor /> <apply> <real /> <ci> a </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> UnitStep </ci> <apply> <imaginary /> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <cn type='integer'> 1 </cn> <ci> a </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LerchPhi", "[", RowBox[List["z_", ",", "1", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "a", ",", RowBox[List["1", "+", "a"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]], ")"]]]], "a"], "+", RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", RowBox[List["2", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ",", "z"]], "]"]]]], RowBox[List["1", "+", "a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Im", "[", "a", "]"]], "]"]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", RowBox[List["Ceiling", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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