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http://functions.wolfram.com/01.19.23.0009.01
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Sum[Sinh[k x]/E^(k a), {k, 1, Infinity}] == E^x/(2 E^a - 2 E^x) +
1/(2 - 2 E^(a + x)) /; Re[a] > Abs[Re[x]]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "k"]], " ", "a"]]], " ", RowBox[List["Sinh", "[", RowBox[List["k", " ", "x"]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", "x"], RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", "a"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", "x"]]]]]], "+", FractionBox["1", RowBox[List["2", "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a", "+", "x"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", RowBox[List["Abs", "[", RowBox[List["Re", "[", "x", "]"]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> a </mi> <mo> + </mo> <mi> x </mi> </mrow> </msup> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <msup> <mi> ⅇ </mi> <mi> x </mi> </msup> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mi> a </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mi> x </mi> </msup> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> a </ci> </apply> </apply> <apply> <sinh /> <apply> <times /> <ci> k </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> a </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <ci> x </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <ci> x </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> a </ci> </apply> <apply> <abs /> <apply> <real /> <ci> x </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "k_"]], " ", "a_"]]], " ", RowBox[List["Sinh", "[", RowBox[List["k_", " ", "x_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", "x"], RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", "a"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", "x"]]]]]], "+", FractionBox["1", RowBox[List["2", "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a", "+", "x"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", RowBox[List["Abs", "[", RowBox[List["Re", "[", "x", "]"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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