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 ArcSin

 http://functions.wolfram.com/01.12.06.0051.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSin", "[", "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[FractionBox[SuperscriptBox["\[Pi]", "2"], "4"], "-", RowBox[List["\[Pi]", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "-", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]], RowBox[List[SuperscriptBox["2", "k"], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]], RowBox[List[SuperscriptBox["2", "k"], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["k", "!"]]]]]]], ")"]], "2"]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "4"], "-", RowBox[List["\[Pi]", " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "+", SuperscriptBox[RowBox[List["ArcCos", "[", "z", "]"]], "2"], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "n"]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SqrtBox["\[Pi]"], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List[FractionBox["3", "2"], "+", "n"]]], RowBox[List["ArcSin", "[", "z", "]"]], RowBox[List["HypergeometricPFQ", "[", 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[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]], "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]], ")"]], "2"], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["3", "+", RowBox[List["2", " ", "n"]]]]], SuperscriptBox[RowBox[List["HypergeometricPFQ", "[", 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[RowBox[List["1", "-", "z"]], "2"]]], "]"]], "2"]]]]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 sin - 1 ( z ) 2 F ( z ) /; ( ( F n ( z ) π 2 4 - π 2 1 - z k = 0 n ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 1 - z ) k 2 k ( 2 k + 1 ) k ! - 2 ( z - 1 ) ( k = 0 n ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 1 - z ) k 2 k ( 2 k + 1 ) k ! ) 2 π 2 4 + cos - 1 ( z ) 2 - π cos - 1 ( z ) + 2 1 2 - n Γ ( n + 3 2 ) ( 2 n + 3 ) π ( n + 1 ) ! ( 1 - z ) n + 3 2 sin - 1 ( z ) 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["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["5", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "z"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 2 - 2 n - 1 Γ ( n + 3 2 ) 2 ( 2 n + 3 ) 2 π ( n + 1 ) ! 2 ( 1 - 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["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["5", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "z"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] 2 ) n ) Condition z 2 Subscript F z Subscript F n z 2 4 -1 -1 2 1 2 1 -1 z 1 2 k 0 n Pochhammer 1 2 k 1 -1 z k 2 k 2 k 1 k -1 -1 2 z -1 k 0 n Pochhammer 1 2 k 1 -1 z k 2 k 2 k 1 k -1 2 2 4 -1 cos -1 z 2 -1 z 2 1 2 -1 n Gamma n 3 2 2 n 3 1 2 n 1 -1 1 -1 z n 3 2 z HypergeometricPFQ 1 n 3 2 n 3 2 n 2 n 5 2 1 -1 z 2 -1 2 -2 n -1 Gamma n 3 2 2 2 n 3 2 n 1 2 -1 1 -1 z 2 n 3 HypergeometricPFQ 1 n 3 2 n 3 2 n 2 n 5 2 1 -1 z 2 -1 2 n [/itex]

 Rule Form

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

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

 2007-05-02