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 LerchPhi

 http://functions.wolfram.com/10.06.03.0063.01

 Input Form

 LerchPhi[z, 5, a] == (1/a^5) (HypergeometricPFQ[{1, a, a, a, a, a}, {1 + a, 1 + a, 1 + a, 1 + a, 1 + a}, z] (1 - 2 UnitStep[-Re[a]])) + (2 ((z HypergeometricPFQ[{1, 1 + a - Ceiling[Re[a]], 1 + a - Ceiling[Re[a]], 1 + a - Ceiling[Re[a]], 1 + a - Ceiling[Re[a]], 1 + a - Ceiling[Re[a]]}, {2 + a - Ceiling[Re[a]], 2 + a - Ceiling[Re[a]], 2 + a - Ceiling[Re[a]], 2 + a - Ceiling[Re[a]], 2 + a - Ceiling[Re[a]]}, z])/(1 + a - Ceiling[Re[a]])^5 + ((1 - Ceiling[Re[a]] + Floor[Re[a]]) UnitStep[Im[a]])/ ((a - Ceiling[Re[a]])^2)^(5/2)) UnitStep[-Re[a]])/z^Ceiling[Re[a]]

 Standard Form

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 MathML Form

 Φ ( z , 5 , a ) TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["5", 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]] 2 ( 1 ( a - Re ( a ) + 1 ) 5 ( z 6 F 5 ( 1 , a - Re ( a ) + 1 , a - Re ( a ) + 1 , a - Re ( a ) + 1 , a - Re ( a ) + 1 , a - Re ( a ) + 1 ; a - Re ( a ) + 2 , a - Re ( a ) + 2 , a - Re ( a ) + 2 , a - Re ( a ) + 2 , a - Re ( a ) + 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "6"], SubscriptBox["F", "5"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftCeiling]", RowBox[List["Re", "(", "a", ")"]], "\[RightCeiling]"]], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] ) + ( - Re ( a ) + Re ( a ) + 1 ) θ UnitStep ( Im ( a ) ) ( ( a - Re ( a ) ) 2 ) 5 / 2 ) θ UnitStep ( - Re ( a ) ) z - Re ( a ) + 1 - 2 θ UnitStep ( - Re ( a ) ) a 5 6 F 5 ( 1 , a , a , a , a , a ; a + 1 , a + 1 , a + 1 , a + 1 , a + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "6"], SubscriptBox["F", "5"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] LerchPhi z 5 a 2 1 a -1 a 1 5 -1 z HypergeometricPFQ 1 a -1 a 1 a -1 a 1 a -1 a 1 a -1 a 1 a -1 a 1 a -1 a 2 a -1 a 2 a -1 a 2 a -1 a 2 a -1 a 2 z -1 a a 1 UnitStep a a -1 a 2 5 2 -1 UnitStep -1 a z -1 a 1 -1 2 UnitStep -1 a a 5 -1 HypergeometricPFQ 1 a a a a a a 1 a 1 a 1 a 1 a 1 z [/itex]

 Rule Form

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

 2007-05-02