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http://functions.wolfram.com/08.05.20.0013.01
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D[EllipticF[z, m], {z, n}] == KroneckerDelta[n] EllipticF[z, m] +
Sum[(1/j!) Sum[Binomial[j, q] Sum[((-1)^q 2^(q - j) Sin[z]^q
(2 p + q - j)^n Binomial[j - q, p]
Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/((j - i - 1)!
(2 Sin[z])^(j - 2 i - 1))) Sum[Binomial[i, s] Pochhammer[1/
2, s] Pochhammer[1/2, i - s] m^(i - s) Cos[z]^(-1 - 2 s)
(1 - m Sin[z]^2)^(-(1/2) - i + s), {s, 0, i}], {i, 0, j - 1}])/
E^((1/2) I ((n - 2 p - q + j) Pi + 2 (2 p + q - j) z)),
{p, 0, j - q}], {q, 0, j - 1}], {j, 1, n}] /;
Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[FractionBox["1", RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["j", ",", "q"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "-", "q"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], SuperscriptBox["2", RowBox[List["q", "-", "j"]]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "q"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "n"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "p"]], "-", "q", "+", "j"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "p"]], "+", "q", "-", "j"]], ")"]], "z"]]]], ")"]]]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "-", "q"]], ",", "p"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["j", "-", "1"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "j"]], ",", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Sin", "[", "z", "]"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", " ", "i"]], "-", "1"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "i"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["i", ",", "s"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "s"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["i", "-", "s"]]]], "]"]], " ", SuperscriptBox["m", RowBox[List["i", "-", "s"]]], SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", "s"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "i", "+", "s"]]]]]]]]], ")"]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> q </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> <mtr> <mtd> <mi> q </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> q </mi> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> q </mi> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> q </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mi> q </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> q </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> j </mi> <mo> - </mo> <mi> q </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> p </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "-", "q"]], Identity, Rule[Editable, True]]], List[TagBox["p", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> i </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "j"]], ")"]], RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> i </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> i </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> s </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> i </mi> </mtd> </mtr> <mtr> <mtd> <mi> s </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> s </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "s"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> i </mi> <mo> - </mo> <mi> s </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["i", "-", "s"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> m </mi> <mrow> <mi> i </mi> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> i </mi> </mrow> <mo> + </mo> <mi> s </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> q </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> j </ci> <ci> q </ci> </apply> <apply> <sum /> <bvar> <ci> p </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <ci> q </ci> </apply> <apply> <power /> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <ci> j </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sin /> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> i </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> s </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> i </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> i </ci> <ci> s </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> s </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <ci> m </ci> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <apply> <cos /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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