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 ArcSinh

 http://functions.wolfram.com/01.25.06.0035.01

 Input Form

 ArcSinh[z] == -((Pi I)/2) + (2 Pi I^Floor[Arg[I (z - x)]/(2 Pi)] Floor[Arg[I (z - x)]/(2 Pi)] + (1/2) Exp[Pi I Floor[Arg[I (z - x)]/(2 Pi)]] (Pi I + 2 ArcSinh[x] + Sqrt[Pi] Sum[(I^(k - 1)/((k - j)! j!)) Pochhammer[-(1/2), -j + k] (1 - I x)^(1/2 + j - k) (1 + I x)^(1/2 - j) Hypergeometric2F1Regularized[1, 1, 3/2 - j, (1/2) (1 + I x)] (z - x)^k, {k, 1, Infinity}, {j, 0, k}])) /; Element[I x, Reals] && I x < -1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "+", 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", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["Exp", "[", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", " ", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", "x", "]"]]]], "+", RowBox[List[SqrtBox["\[Pi]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[SuperscriptBox["\[ImaginaryI]", RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "j"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "-", "j"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ")"]]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]]]], ")"]]]]]], ")"]]]]]], " ", "/;", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "<", RowBox[List["-", "1"]]]]]]]]]]

 MathML Form

 sinh - 1 ( z ) - π 2 + ( 2 π arg ( ( z - x ) ) 2 π arg ( ( z - x ) ) 2 π + 1 2 π arg ( ( z - x ) ) 2 π ( π + 2 sinh - 1 ( x ) + π k = 1 j = 0 k k - 1 ( - 1 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( k - j ) ! j ! ( 1 - x ) j - k + 1 2 ( x + 1 ) 1 2 - j 2 F ~ 1 ( 1 , 1 ; 3 2 - j ; 1 2 ( x + 1 ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["3", "2"], "-", "j"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]], ")"]]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ( z - x ) k ) ) /; x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] x < - 1 Condition z -1 2 -1 2 z -1 x 2 -1 z -1 x 2 -1 1 2 z -1 x 2 -1 2 x 1 2 j 0 k k 1 k -1 Pochhammer -1 1 2 k -1 j k -1 j j -1 1 -1 x j -1 k 1 2 x 1 1 2 -1 j Hypergeometric2F1Regularized 1 1 3 2 -1 j 1 2 x 1 z -1 x k x x -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]], "+", 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", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["2", " ", RowBox[List["ArcSinh", "[", "x", "]"]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["k", "-", "1"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "j"]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "1", ",", RowBox[List[FractionBox["3", "2"], "-", "j"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]]]]]]]]]]]]], ")"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "<", RowBox[List["-", "1"]]]]]]]]]]]]

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

 2007-05-02