Hyperbolic sine: Transformations (formula 01.19.16.0035)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Transformations

Transformations and argument simplifications

Argument involving inverse trigonometric and hyperbolic functions

Involving tan^-1

http://functions.wolfram.com/01.19.16.0035.01

Input Form

Sinh[I n ArcTan[z]] == ((I z)/Sqrt[1 + z^2]) ChebyshevU[n - 1, 1/Sqrt[1 + z^2]] /; Element[n, Integers] && n > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["\[ImaginaryI]", " ", "n", " ", RowBox[List["ArcTan", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

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> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> <mtext> </mtext> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> U </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sinh /> <apply> <times /> <imaginaryi /> <ci> n </ci> <apply> <arctan /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ChebyshevU </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sinh", "[", RowBox[List["\[ImaginaryI]", " ", "n_", " ", RowBox[List["ArcTan", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

2001-10-29