ArcSin[z] == (z/(2 Sqrt[Pi])) (1/(2 Pi I)) Integrate[(Gamma[s] Gamma[1/2 - s]^2)/Gamma[3/2 - s]/(-z^2)^s, {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; 0 < \[Gamma] < 1/2 && Abs[Arg[-z^2]] < Pi

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["z", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], FractionBox["1", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], " "]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], "2"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "s"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]", "<", FractionBox["1", "2"]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> z </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mi> &#947; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mrow> <mi> &#947; </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </msubsup> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mtext> </mtext> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> &#947; </mi> <mo> &lt; </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsin /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> s </ci> </bvar> <lowlimit> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> &#947; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <lt /> <apply> <abs /> <apply> <arg /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <pi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "s"]], "]"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]", "<", FractionBox["1", "2"]]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]]]

2001-10-29