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AppellF1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > AppellF1[a,b1,b2,c,z1,z2] > Specific values > Values at fixed points > For fixed z1,z2





http://functions.wolfram.com/07.36.03.0014.01









  


  








Input Form





AppellF1[n + 1/2, n + 1, 1, n + 3/2, Subscript[z, 1], Subscript[z, 2]] == (-(-1)^n) (1 + 2 n) ArcTanh[Sqrt[Subscript[z, 2]]] (Subscript[z, 1] - Subscript[z, 2])^(-1 - n) Sqrt[Subscript[z, 2]] + ((2 n + 1)/(2 n!)) Sum[(((-1)^(n - m) Pochhammer[m, 2 (n - m)])/ ((n - m)! (2 Sqrt[Subscript[z, 1]])^(2 n - m))) ((-1)^m m! ((-Sqrt[Subscript[z, 2]] + Sqrt[Subscript[z, 1]])^(-1 - m) + (Sqrt[Subscript[z, 2]] + Sqrt[Subscript[z, 1]])^(-1 - m)) ArcTanh[Sqrt[Subscript[z, 1]]] + Sum[Binomial[m, j] (-1)^(m - j) (m - j)! (1/(Sqrt[Subscript[z, 1]] - Sqrt[Subscript[z, 2]])^ (m - j + 1) + 1/(Sqrt[Subscript[z, 1]] + Sqrt[Subscript[z, 2]])^ (m - j + 1)) Sum[((k! Pochhammer[2 k - j + 2, 2 (j - k - 1)])/ ((j - k - 1)! (2 Sqrt[Subscript[z, 1]])^(j - 2 k - 1))) (1 - Subscript[z, 1])^(-k - 1), {k, 0, j - 1}], {j, 0, m}]), {m, 0, n}] /; Element[n, Integers] && n >= 0








Standard Form





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MathML Form







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-1 </cn> <ci> j </ci> </apply> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </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> <factorial /> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> 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type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctanh /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02