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; }

 ArcSech

 http://functions.wolfram.com/01.30.06.0109.01

 Input Form

 ArcSech[z]^2 == Subscript[F, Infinity][z] /; Subscript[F, n][z] == -(Pi^2/4) + (Pi/z) Sum[Pochhammer[1/2, k]/(z^(2 k) ((2 k + 1) k!)), {k, 0, n}] - (1/z^2) Sum[(2^(2 k) k!^2)/(z^(2 k) ((2 k + 1)! (k + 1))), {k, 0, n}] == ArcSech[z]^2 - (Sqrt[Pi]/2) z^(-3 - 2 n) Gamma[3/2 + n]^2 HypergeometricPFQRegularized[{1, 3/2 + n, 3/2 + n}, {2 + n, 5/2 + n}, 1/z^2] + (Sqrt[Pi]/2) z^(-4 - 2 n) Gamma[2 + n]^2 HypergeometricPFQRegularized[{1, 2 + n, 2 + n}, {5/2 + n, 3 + n}, 1/z^2] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSech", "[", "z", "]"]], "2"], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"]]], "+", RowBox[List[FractionBox["\[Pi]", "z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[FractionBox["1", SuperscriptBox["z", "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox[RowBox[List["k", "!"]], "2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["ArcSech", "[", "z", "]"]], "2"], StyleBox["-", Rule[FontWeight, "Plain"]], RowBox[List[StyleBox[FractionBox[SqrtBox["\[Pi]"], "2"], Rule[FontWeight, "Plain"]], StyleBox[SuperscriptBox["z", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", "n"]]]]], Rule[FontWeight, "Plain"]], StyleBox[" ", Rule[FontWeight, "Plain"]], StyleBox[SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], "2"], Rule[FontWeight, "Plain"]], StyleBox[" ", Rule[FontWeight, "Plain"]], RowBox[List[StyleBox["HypergeometricPFQRegularized", Rule[FontWeight, "Plain"]], StyleBox["[", Rule[FontWeight, "Plain"]], RowBox[List[StyleBox[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], Rule[FontWeight, "Plain"]], StyleBox[",", Rule[FontWeight, "Plain"]], StyleBox[RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], Rule[FontWeight, "Plain"]], StyleBox[",", Rule[FontWeight, "Plain"]], FractionBox[StyleBox["1", Rule[FontWeight, "Plain"]], SuperscriptBox[StyleBox["z", Rule[FontWeight, "Plain"]], "2"]]]], StyleBox["]", Rule[FontWeight, "Plain"]]]]]], "+", RowBox[List[FractionBox[SqrtBox["\[Pi]"], "2"], SuperscriptBox["z", RowBox[List[RowBox[List["-", "4"]], "-", RowBox[List["2", "n"]]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "+", "n"]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["5", "2"], "+", "n"]], ",", RowBox[List["3", "+", "n"]]]], "}"]], ",", FractionBox["1", SuperscriptBox[StyleBox["z", Rule[FontWeight, "Plain"]], "2"]]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 sech - 1 ( z ) 2 F ( z ) /; ( ( F n ( z ) - 1 z 2 k = 0 n 2 2 k k ! 2 z - 2 k ( 2 k + 1 ) ! ( k + 1 ) + π z k = 0 n ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] z - 2 k ( 2 k + 1 ) k ! - π 2 4 - 1 2 π Γ ( n + 3 2 ) 2 z - 2 n - 3 3 F ~ 2 ( 1 , n + 3 2 , n + 3 2 ; n + 2 , n + 5 2 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["5", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox[StyleBox["z", Rule[FontWeight, "Plain"]], "2"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + 1 2 π Γ ( n + 2 ) 2 z - 2 n - 4 3 F ~ 2 ( 1 , n + 2 , n + 2 ; n + 5 2 , n + 3 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", FractionBox["5", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "3"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox[StyleBox["z", Rule[FontWeight, "Plain"]], "2"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + sech - 1 ( z ) 2 ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) Condition z 2 Subscript F z Subscript F n z -1 1 z 2 -1 k 0 n 2 2 k k 2 z -2 k 2 k 1 k 1 -1 z -1 k 0 n Pochhammer 1 2 k z -2 k 2 k 1 k -1 -1 2 4 -1 -1 1 2 1 2 Gamma n 3 2 2 z -2 n -3 HypergeometricPFQRegularized 1 n 3 2 n 3 2 n 2 n 5 2 1 z 2 -1 1 2 1 2 Gamma n 2 2 z -2 n -4 HypergeometricPFQRegularized 1 n 2 n 2 n 5 2 n 3 1 z 2 -1 z 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcSech", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["\[Pi]", "2"], "4"]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "z"], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]]]]]]], SuperscriptBox["z", "2"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["ArcSech", "[", "z", "]"]], "2"], "-", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List[FractionBox["3", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "4"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "+", "n"]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["5", "2"], "+", "n"]], ",", RowBox[List["3", "+", "n"]]]], "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02

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