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EllipticF




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > General characteristics > Branch cuts > With respect to z > General description





http://functions.wolfram.com/08.05.04.0013.01









  


  








Input Form





BranchCuts[EllipticF[z, m], z] == {SequenceList[{Interval[{ArcCsc[Sqrt[m]] + Pi k, Pi/2 + Pi k}], I}, Element[k, Integers] && Element[m, Reals] && m > 1], SequenceList[{Interval[{Pi/2 + Pi k, -ArcCsc[Sqrt[m]] + Pi (k + 1)}], -I}, Element[k, Integers] && Element[m, Reals] && m > 1], SequenceList[{Interval[{Pi/2 + 2 Pi k, Pi/2 + 2 Pi k + I Infinity}], 1}, Element[k, Integers] && !(Element[m, Reals] && 0 < m < 1)] || SequenceList[{Interval[{Pi - ArcCsc[Sqrt[m]] + 2 Pi k, Pi/2 + 2 Pi k + I Infinity}], 1}, Element[k, Integers] && Element[m, Reals] && 0 < m < 1], SequenceList[{Interval[{(3 Pi)/2 + 2 Pi k, (3 Pi)/2 + 2 Pi k + I Infinity}], -1}, Element[k, Integers] && !(Element[m, Reals] && 0 < m < 1)] || SequenceList[{Interval[{2 Pi - ArcCsc[Sqrt[m]] + 2 Pi k, (3 Pi)/2 + 2 Pi k + I Infinity}], -1}, Element[k, Integers] && Element[m, Reals] && 0 < m < 1], SequenceList[{Interval[{Pi/2 + 2 Pi k - I Infinity, Pi/2 + 2 Pi k}], 1}, Element[k, Integers] && !(Element[m, Reals] && 0 < m < 1)] || SequenceList[{Interval[{Pi/2 + 2 Pi k - I Infinity, 2 Pi k + ArcCsc[Sqrt[m]]}], 1}, Element[k, Integers] && Element[m, Reals] && 0 < m < 1], SequenceList[{Interval[{(3 Pi)/2 + 2 Pi k - I Infinity, (3 Pi)/2 + 2 Pi k}], -1}, Element[k, Integers] && !(Element[m, Reals] && 0 < m < 1)] || SequenceList[{Interval[{(3 Pi)/2 + 2 Pi k - I Infinity, 2 Pi k + Pi + ArcCsc[Sqrt[m]]}], -1}, Element[k, Integers] && Element[m, Reals] && 0 < m < 1]}








Standard Form





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MathML Form







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</mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> &#960; </mi> <mo> + </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> &#8492;&#119966; </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mi> &#8520; </mi> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> &#960; </mi> <mo> - </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> &#8744; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> &#960; </mi> <mo> + </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> m </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02