html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Sinh

 http://functions.wolfram.com/01.19.06.0062.01

 Input Form

 Sinh[z]^n == Subscript[F, Infinity][z] /; Subscript[F, m][z] == I^n (1 + 2^(1 - n) Sum[((1/(2 j)!) Sum[Binomial[n, k] (n - 2 k)^(2 j), {k, 0, Floor[(n - 1)/2]}]) (z - (Pi I)/2)^(2 j), {j, 1, m}]) == Sinh[z]^n - ((I^n Sqrt[Pi])/2^(n + 2 m + 1)) (z - (Pi I)/2)^(2 + 2 m) Sum[Binomial[n, k] (n - 2 k)^(2 + 2 m) HypergeometricPFQRegularized[{1}, {3/2 + m, 2 + m}, ((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["Sinh", "[", "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["\[ImaginaryI]", "n"], RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "m"], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "j"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["2", "j"]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["2", "j"]]]]]]]]]]], ")"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "n"], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], SqrtBox["\[Pi]"]]], SuperscriptBox["2", RowBox[List["n", "+", RowBox[List["2", "m"]], "+", "1"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["2", "+", RowBox[List["2", "m"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["2", "+", RowBox[List["2", "m"]]]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "m"]], ",", RowBox[List["2", "+", "m"]]]], "}"]], ",", 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

 sinh n ( z ) F ( z ) /; ( ( F m ( z ) n ( 1 + 2 1 - n j = 1 m 1 ( 2 j ) ! ( ( - 1 ) j k = 0 n - 1 2 ( 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 ) ( z - π 2 ) 2 j ) sinh n ( z ) - n π 2 n + 2 m + 1 ( z - π 2 ) 2 + 2 m k = 0 n - 1 2 ( 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 m + 2 1 F ~ 2 ( 1 ; m + 3 2 , m + 2 ; ( n - 2 k ) 2 4 ( z - π 2 ) 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["m", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, 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"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) m n + ) Condition z n Subscript F z Subscript F m z n 1 2 1 -1 n j 1 m 1 2 j -1 -1 j k 0 n -1 2 -1 Binomial n k n -1 2 k 2 j z -1 2 -1 2 j z n -1 n 1 2 2 n 2 m 1 -1 z -1 2 -1 2 2 m k 0 n -1 2 -1 Binomial n k n -1 2 k 2 m 2 HypergeometricPFQRegularized 1 m 3 2 m 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["Sinh", "[", "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["\[ImaginaryI]", "n"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "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[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], 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["2", " ", "j"]], ")"]], "!"]]]]]]]]], ")"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "n"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], " ", SqrtBox["\[Pi]"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], RowBox[List["2", "+", RowBox[List["2", " ", "m"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["2", "+", RowBox[List["2", " ", "m"]]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "m"]], ",", RowBox[List["2", "+", "m"]]]], "}"]], ",", 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"]]]]], "]"]]]]]]]], SuperscriptBox["2", RowBox[List["n", "+", RowBox[List["2", " ", "m"]], "+", "1"]]]]]]]], "&&", 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