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Power




Mathematica Notation

Traditional Notation









Elementary Functions > Power[z,a] > Products > Regularized infinite products





http://functions.wolfram.com/01.02.24.0008.01









  


  








Input Form





Product[k^4 + z^4, {k, 0, Infinity}] == 2 z^2 (Cosh[Sqrt[2] Pi z] - Cos[Sqrt[2] Pi z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["(", RowBox[List[SuperscriptBox["k", "4"], "+", SuperscriptBox["z", "4"]]], ")"]]]], "\[Equal]", RowBox[List["2", SuperscriptBox["z", "2"], RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[SqrtBox["2"], "\[Pi]", " ", "z"]], "]"]], " ", "-", RowBox[List["Cos", "[", RowBox[List[SqrtBox["2"], "\[Pi]", " ", "z"]], "]"]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <msup> <mi> k </mi> <mn> 4 </mn> </msup> <mo> + </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <power /> <ci> k </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <cosh /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], RowBox[List["(", RowBox[List[SuperscriptBox["k_", "4"], "+", SuperscriptBox["z_", "4"]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List["2", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[SqrtBox["2"], " ", "\[Pi]", " ", "z"]], "]"]], "-", RowBox[List["Cos", "[", RowBox[List[SqrtBox["2"], " ", "\[Pi]", " ", "z"]], "]"]]]], ")"]]]]]]]]


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





2002-12-18