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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > Series representations > Generalized power series > Expansions on branch cuts > Formulas for vertical intervals > For Re(z0/2 Pi-1/4) ∈ Z





http://functions.wolfram.com/08.05.06.0028.01









  


  










Input Form





EllipticF[z, m] == (2 (2 Re[Subscript[z, 0]/(2 Pi) - 1/4] + 1) EllipticK[m] - (1/Sqrt[m]) EllipticK[1/m]) (1 + Exp[(-Pi) I (Floor[3/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[3/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])]) - Exp[(-Pi) I (Floor[3/4 + Arg[z - Subscript[z, 0]]/(2 Pi)] + Floor[3/4 - Arg[z - Subscript[z, 0]]/(2 Pi)])] EllipticF[Subscript[z, 0], m] + Sum[(1/k!) Sum[(1/j!) Sum[Binomial[j, q] Sum[((-1)^q 2^(q - j) Sin[Subscript[z, 0]]^q (2 p + q - j)^k Binomial[j - q, p] Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/( (j - i - 1)! (2 Sin[Subscript[z, 0]])^(j - 2 i - 1))) Sum[Binomial[i, s] Pochhammer[1/2, s] Pochhammer[1/2, i - s] m^(i - s) Cos[Subscript[z, 0]]^(-1 - 2 s) (1 - m Sin[Subscript[z, 0]]^2)^(-(1/2) - i + s), {s, 0, i}], {i, 0, j - 1}])/E^((1/2) I ((k - 2 p - q + j) Pi + 2 (2 p + q - j) Subscript[z, 0])), {p, 0, j - q}], {q, 0, j - 1}], {j, 1, k}] (z - Subscript[z, 0])^k, {k, 1, Infinity}] /; Element[Re[Subscript[z, 0]/(2 Pi) - 1/4], Integers]










Standard Form





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MathML Form







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-1 </cn> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <apply> <plus /> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z 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<sep /> 4 </cn> </apply> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> EllipticF </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> 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Rule Form





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





2007-05-02