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http://functions.wolfram.com/01.06.21.1620.01
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Integrate[1/Sqrt[1 - m Sin[t]^2], {t, k Pi + ArcCsc[Sqrt[m]],
k Pi + Pi/2}] == (2 k + 1) EllipticK[m] -
EllipticF[k Pi + ArcCsc[Sqrt[m]], m] /; Element[k, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List[RowBox[List["k", " ", "\[Pi]"]], "+", RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]]]], RowBox[List[RowBox[List["k", " ", "\[Pi]"]], "+", FractionBox["\[Pi]", "2"]]]], RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "t", "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[RowBox[List["k", " ", "\[Pi]"]], "+", RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]]]], ",", "m"]], "]"]]]]]], "/;", RowBox[List["k", "\[Element]", "Integers"]]]]]]
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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> <msubsup> <mo> ∫ </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> π </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> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> π </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> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <plus /> <apply> <times /> <ci> k </ci> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> n </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> t </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <apply> <plus /> <apply> <times /> <ci> k </ci> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List[RowBox[List["k_", " ", "\[Pi]"]], "+", RowBox[List["ArcCsc", "[", SqrtBox["m_"], "]"]]]], RowBox[List[RowBox[List["k_", " ", "\[Pi]"]], "+", FractionBox["\[Pi]", "2"]]]], RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "-", RowBox[List["m_", " ", SuperscriptBox[RowBox[List["Sin", "[", "t_", "]"]], "2"]]]]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "-", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[RowBox[List["k", " ", "\[Pi]"]], "+", RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]]]], ",", "m"]], "]"]]]], "/;", RowBox[List["k", "\[Element]", "Integers"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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