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http://functions.wolfram.com/01.19.23.0008.01
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Sum[z^k Sinh[k a], {k, 1, n}] ==
(z ((z Sinh[a n] - Sinh[a (n + 1)]) z^n + Sinh[a]))/(z^2 - 2 Cosh[a] z + 1)
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[SuperscriptBox["z", "k"], " ", RowBox[List["Sinh", "[", RowBox[List["k", " ", "a"]], "]"]]]]]], "\[Equal]", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "n"]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List["a", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox["z", "n"]]], "+", RowBox[List["Sinh", "[", "a", "]"]]]], ")"]]]], RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["2", " ", RowBox[List["Cosh", "[", "a", "]"]], " ", "z"]], "+", "1"]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> <mo> + </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <sinh /> <apply> <times /> <ci> k </ci> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <sinh /> <apply> <times /> <ci> a </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sinh /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <apply> <sinh /> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cosh /> <ci> a </ci> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </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"]], "n_"], RowBox[List[SuperscriptBox["z_", "k_"], " ", RowBox[List["Sinh", "[", RowBox[List["k_", " ", "a_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Sinh", "[", RowBox[List["a", " ", "n"]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List["a", " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox["z", "n"]]], "+", RowBox[List["Sinh", "[", "a", "]"]]]], ")"]]]], RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["2", " ", RowBox[List["Cosh", "[", "a", "]"]], " ", "z"]], "+", "1"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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