Incomplete elliptic integral of the first kind: General characteristics (formula 08.05.04.0018)

EllipticF

$Mathematica Notation$

Elliptic Integrals

EllipticF[z,m]

General characteristics

Branch cuts

With respect to z

Formulas for vertical intervals

For m>0

http://functions.wolfram.com/08.05.04.0018.01

Input Form

Limit[EllipticF[Pi/2 + 2 k Pi + I x - \[Epsilon], m], \[Epsilon] -> Plus[0]] == EllipticF[Pi/2 + 2 k Pi + I x, m] /; Element[x, Reals] && Element[k, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["2", "k", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "x"]], "-", "\[Epsilon]"]], ",", "m"]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List["2", "k", " ", "\[Pi]"]], "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], ",", "m"]], "]"]]]], "/;", " ", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> ϵ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> ϵ </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> ϵ </ci> </bvar> <condition> <apply> <tendsto /> <ci> ϵ </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> EllipticF </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> </apply> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ϵ </ci> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <ci> EllipticF </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> </apply> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02