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http://functions.wolfram.com/01.19.24.0002.01
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Product[Sinh[z + (2 Pi k I)/n], {k, 1, n - 1}] ==
((-2)^(1 - n) (-Cos[(n Pi)/2] + Cos[(n Pi)/2 - I n z]) Csch[z])/I^n /;
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List["Sinh", "[", RowBox[List["z", "+", FractionBox[RowBox[List["2", "\[Pi]", " ", "k", " ", "\[ImaginaryI]"]], "n"]]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], RowBox[List["1", "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]]]], "+", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", "n", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Csch", "[", "z", "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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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> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> ⅈ </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <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> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <sinh /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <ci> k </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> -2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <cos /> <apply> <plus /> <apply> <times /> <ci> n </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> n </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <ci> n </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], RowBox[List["n_", "-", "1"]]], RowBox[List["Sinh", "[", RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "k_", " ", "\[ImaginaryI]"]], "n_"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], RowBox[List["1", "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]]]], "+", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", "n", " ", "z"]]]], "]"]]]], ")"]], " ", RowBox[List["Csch", "[", "z", "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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