Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ArcSinh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Complex characteristics > Real part





http://functions.wolfram.com/01.25.19.0001.01









  


  










Input Form





Re[ArcSinh[x + I y]] == (1/2) Log[X^2 + Y^2] /; X == x + Sqrt[(1/2) (x^2 - y^2 + Sqrt[(x^2 + y^2 + 1)^2 - 4 y^2] + 1)] && Y == y + Sign[x y] Sqrt[(1/2) (-x^2 + y^2 + Sqrt[(x^2 + y^2 + 1)^2 - 4 y^2] - 1)] && !IntervalMemberQ[Interval[{-Infinity, -1}], I (x + I y)] && !IntervalMemberQ[Interval[{1, Infinity}], I (x + I y)]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Re", "[", RowBox[List["ArcSinh", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["X", "2"], "+", SuperscriptBox["Y", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["X", "\[Equal]", RowBox[List["x", "+", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "2"], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", "1"]], ")"]], "2"], "-", RowBox[List["4", " ", SuperscriptBox["y", "2"]]]]]], "+", "1"]], ")"]]]]]]]]], "\[And]", RowBox[List["Y", "\[Equal]", RowBox[List["y", "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["x", " ", "y"]], "]"]], " ", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["x", "2"]]], "+", SuperscriptBox["y", "2"], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", "1"]], ")"]], "2"], "-", RowBox[List["4", " ", SuperscriptBox["y", "2"]]]]]], "-", "1"]], ")"]]]]]]]]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]]], "]"]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> X </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> Y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> X </mi> <mo> &#10869; </mo> <mrow> <mi> x </mi> <mo> + </mo> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> Y </mi> <mo> &#10869; </mo> <mrow> <mi> y </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> X </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> Y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> X </mi> <mo> &#10869; </mo> <mrow> <mi> x </mi> <mo> + </mo> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> Y </mi> <mo> &#10869; </mo> <mrow> <mi> y </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msqrt> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", RowBox[List["ArcSinh", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox["X", "2"], "+", SuperscriptBox["Y", "2"]]], "]"]]]], "/;", RowBox[List[RowBox[List["X", "\[Equal]", RowBox[List["x", "+", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "2"], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", "1"]], ")"]], "2"], "-", RowBox[List["4", " ", SuperscriptBox["y", "2"]]]]]], "+", "1"]], ")"]]]]]]]]], "&&", RowBox[List["Y", "\[Equal]", RowBox[List["y", "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["x", " ", "y"]], "]"]], " ", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["x", "2"]]], "+", SuperscriptBox["y", "2"], "+", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", "1"]], ")"]], "2"], "-", RowBox[List["4", " ", SuperscriptBox["y", "2"]]]]]], "-", "1"]], ")"]]]]]]]]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], ")"]]]]]], "]"]]]]]]]]]]]]










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





2001-10-29