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 ArcSinh

 http://functions.wolfram.com/01.25.06.0026.01

 Input Form

 ArcSinh[z] == -((Pi I)/2) + (1/(1 - I Subscript[z, 0]))^ ((1/2) Floor[Arg[I (Subscript[z, 0] - z)]/(2 Pi)]) (1 - I Subscript[z, 0])^((1/2) Floor[Arg[I (Subscript[z, 0] - z)]/(2 Pi)]) (2 Pi I^Floor[Arg[I (z - Subscript[z, 0])]/(2 Pi)] Floor[Arg[I (z - Subscript[z, 0])]/(2 Pi)] Floor[(Pi + Arg[1 + I Subscript[z, 0]])/(2 Pi)] + (1/2) (1/(1 + I Subscript[z, 0]))^ ((1/2) Floor[Arg[I (z - Subscript[z, 0])]/(2 Pi)]) (1 + I Subscript[z, 0])^((1/2) Floor[Arg[I (z - Subscript[z, 0])]/ (2 Pi)]) (Pi I + Sqrt[Pi] Sum[((2^k Subscript[z, 0]^(1 - k))/k!) HypergeometricPFQRegularized[{1/2, 1/2, 1}, {1 - k/2, (3 - k)/2}, -Subscript[z, 0]^2] (z - Subscript[z, 0])^k, {k, 0, Infinity}]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "-", "z"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List[SubscriptBox["z", "0"], "-", "z"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", "\[Pi]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["z", "0"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "k"], SubsuperscriptBox["z", "0", RowBox[List["1", "-", " ", "k"]]]]], RowBox[List["k", "!"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["k", "2"]]], ",", FractionBox[RowBox[List["3", "-", "k"]], "2"]]], "}"]], ",", RowBox[List["-", SubsuperscriptBox["z", "0", "2"]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]], ")"]]]]]], ")"]], " "]]]]]]]]

 MathML Form

 sinh - 1 ( z ) - π 2 + ( 1 1 - z 0 ) 1 2 arg ( ( z 0 - z ) ) 2 π ( 1 - z 0 ) 1 2 arg ( ( z 0 - z ) ) 2 π ( 2 π arg ( ( z - z 0 ) ) 2 π arg ( ( z - z 0 ) ) 2 π arg ( z 0 + 1 ) + π 2 π + 1 2 ( 1 z 0 + 1 ) 1 2 arg ( ( z - z 0 ) ) 2 π ( z 0 + 1 ) 1 2 arg ( ( z - z 0 ) ) 2 π ( π + π k = 0 2 k z 0 1 - k k ! 3 F ~ 2 ( 1 2 , 1 2 , 1 ; 1 - k 2 , 3 - k 2 ; - z 0 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[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["k", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "k"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SubsuperscriptBox["z", "0", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( z - z 0 ) k ) ) z -1 2 -1 1 1 -1 Subscript z 0 -1 1 2 Subscript z 0 -1 z 2 -1 1 -1 Subscript z 0 1 2 Subscript z 0 -1 z 2 -1 2 z -1 Subscript z 0 2 -1 z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 1 2 1 Subscript z 0 1 -1 1 2 z -1 Subscript z 0 2 -1 Subscript z 0 1 1 2 z -1 Subscript z 0 2 -1 1 2 k 0 2 k Subscript z 0 1 -1 k k -1 HypergeometricPFQRegularized 1 2 1 2 1 1 -1 k 2 -1 3 -1 k 2 -1 -1 Subscript z 0 2 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SubscriptBox["zz", "0"], "-", "z"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SubscriptBox["zz", "0"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "k"], " ", SubsuperscriptBox["zz", "0", RowBox[List["1", "-", "k"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["k", "2"]]], ",", FractionBox[RowBox[List["3", "-", "k"]], "2"]]], "}"]], ",", RowBox[List["-", SubsuperscriptBox["zz", "0", "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]]]], ")"]]]]]]]]]]

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

 2007-05-02