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 Cosh

 http://functions.wolfram.com/01.20.06.0062.01

 Input Form

 Cosh[z]^n == Subscript[F, Infinity][z] /; Subscript[F, m][z] == 2^(1 - n) I^n (z - (Pi I)/2)^n Sum[((1/(n + 2 j)!) Sum[(-1)^k Binomial[n, k] (n - 2 k)^(n + 2 j), {k, 0, Floor[(n - 1)/2]}]) (z - (Pi I)/2)^(2 j), {j, 0, m}] == 2^(1 - n) I^n (z - (Pi I)/2)^n Sum[(-1)^k Binomial[n, k] (-2 k + n)^n ((1/n!) HypergeometricPFQ[{1}, {1/2 + n/2, 1 + n/2}, ((n - 2 k)^2/4) (z - (Pi I)/2)^2] - (((n - 2 k)^2 (z - (Pi I)/2)^2)^(1 + m)/Gamma[3 + 2 m + n]) HypergeometricPFQ[{1}, {m + (n + 3)/2, 2 + m + n/2}, ((n - 2 k)^2/4) (z - (Pi I)/2)^2]), {k, 0, Floor[(n - 1)/2]}] && Element[m, Integers] && m >= 0 && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "n"], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], SuperscriptBox["\[ImaginaryI]", "n"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "+", RowBox[List["2", " ", "j"]]]], ")"]], "!"]]], " ", 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"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", RowBox[List["2", " ", "j"]]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["2", "j"]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], SuperscriptBox["\[ImaginaryI]", "n"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "n"], 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"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["n", "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["n", "2"]]], ",", RowBox[List["1", "+", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], "4"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]], ")"]], RowBox[List["1", "+", "m"]]], RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", "n"]], "]"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", FractionBox[RowBox[List["n", "+", "3"]], "2"]]], ",", RowBox[List["2", "+", "m", "+", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], "4"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]]]], "]"]]]]]], ")"]]]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]], ")"]]]]]]

 MathML Form

 cosh n ( z ) F ( z ) /; ( ( F m ( z ) 2 1 - n n ( z - π 2 ) n j = 0 m ( 1 ( 2 j + n ) ! k = 0 n - 1 2 ( - 1 ) k ( 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]] ( n - 2 k ) 2 j + n ) ( z - π 2 ) 2 j 2 1 - n n ( z - π 2 ) n k = 0 n - 1 2 ( - 1 ) k ( 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]] ( n - 2 k ) n ( 1 n ! 1 F 2 ( 1 ; n 2 + 1 2 , n 2 + 1 ; ( n - 2 k ) 2 4 ( z - π 2 ) 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["n", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["n", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], "4"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - ( ( n - 2 k ) 2 ( z - π 2 ) 2 ) m + 1 Γ ( 2 m + n + 3 ) 1 F 2 ( 1 ; n + 3 2 + m , m + n 2 + 2 ; ( n - 2 k ) 2 4 ( z - π 2 ) 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["n", "+", "3"]], "2"], "+", "m"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["n", "2"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], "4"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) m n + ) Condition z n Subscript F z Subscript F m z 2 1 -1 n n z -1 2 -1 n j 0 m 1 2 j n -1 k 0 n -1 2 -1 -1 k Binomial n k n -1 2 k 2 j n z -1 2 -1 2 j 2 1 -1 n n z -1 2 -1 n k 0 n -1 2 -1 -1 k Binomial n k n -1 2 k n 1 n -1 HypergeometricPFQ 1 n 2 -1 1 2 n 2 -1 1 n -1 2 k 2 4 -1 z -1 2 -1 2 -1 n -1 2 k 2 z -1 2 -1 2 m 1 Gamma 2 m n 3 -1 HypergeometricPFQ 1 n 3 2 -1 m m n 2 -1 2 n -1 2 k 2 4 -1 z -1 2 -1 2 m n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["Cosh", "[", "z_", "]"]], "n_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["\[ImaginaryI]", "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", 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"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", RowBox[List["2", " ", "j"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["2", " ", "j"]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", RowBox[List["2", " ", "j"]]]], ")"]], "!"]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["\[ImaginaryI]", "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "n"], " ", 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"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", FractionBox["n", "2"]]], ",", RowBox[List["1", "+", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]]]], "]"]], RowBox[List["n", "!"]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]], ")"]], RowBox[List["1", "+", "m"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["m", "+", FractionBox[RowBox[List["n", "+", "3"]], "2"]]], ",", RowBox[List["2", "+", "m", "+", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], "2"]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", "n"]], "]"]]]]], ")"]]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02