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 Sinh

 http://functions.wolfram.com/01.19.20.0010.01

 Input Form

 D[Sinh[z]^n, {z, \[Alpha]}] == (I^n Binomial[n, n/2] (1 - Mod[n, 2]) (1/(z^\[Alpha] Gamma[1 - \[Alpha]])))/ 2^n - (2^(\[Alpha] - n) I^n Sqrt[Pi] Sum[(-1)^(k - 1) Binomial[n, k] (2 Cos[(n Pi)/2] HypergeometricPFQRegularized[{1}, {1/2 - \[Alpha]/2, 1 - \[Alpha]/2}, (1/4) (-2 k + n)^2 z^2] - I (-2 k + n) z HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, 3/2 - \[Alpha]/2}, (1/4) (-2 k + n)^2 z^2] Sin[(n Pi)/2]), {k, 0, Floor[(n - 1)/2]}])/z^\[Alpha] /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "n"]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", SuperscriptBox["\[ImaginaryI]", "n"], RowBox[List["Binomial", "[", RowBox[List["n", ",", FractionBox["n", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]]]], ")"]], FractionBox[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "n"]]], SuperscriptBox["\[ImaginaryI]", "n"], SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], " ", "z", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 α sinh n ( z ) z α 2 - n n ( 1 - n mod 2 \$CellContext`n 2 ) z - α Γ ( 1 - α ) ( n n 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["n", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] - 2 α - n n π z - α k = 0 n - 1 2 ( - 1 ) k - 1 ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 cos ( n π 2 ) 1 F ~ 2 ( 1 ; 1 2 - α 2 , 1 - α 2 ; 1 4 ( n - 2 k ) 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - ( n - 2 k ) z 1 F ~ 2 ( 1 ; 1 - α 2 , 3 2 - α 2 ; 1 4 ( n - 2 k ) 2 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] sin ( n π 2 ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + Condition z α z n 2 -1 n n 1 -1 \$CellContext`n 2 z -1 α Gamma 1 -1 α -1 Binomial n n 2 -1 -1 2 α -1 n n 1 2 z -1 α k 0 n -1 2 -1 -1 k -1 Binomial n k 2 n 2 -1 HypergeometricPFQRegularized 1 1 2 -1 α 2 -1 1 -1 α 2 -1 1 4 n -1 2 k 2 z 2 -1 n -1 2 k z HypergeometricPFQRegularized 1 1 -1 α 2 -1 3 2 -1 α 2 -1 1 4 n -1 2 k 2 z 2 n 2 -1 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], SuperscriptBox[RowBox[List["Sinh", "[", "z_", "]"]], "n_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", SuperscriptBox["\[ImaginaryI]", "n"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", FractionBox["n", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "n"]]], " ", SuperscriptBox["\[ImaginaryI]", "n"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], " ", "z", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2002-12-18