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variants of this functions
Hypergeometric2F1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Specific values > For rational parameters with denominators 5 and fixed z > For fixed z and a=5, b>=a > For fixed z and a=5, b=5





http://functions.wolfram.com/07.23.03.b488.01









  


  








Input Form





Hypergeometric2F1[5, 5, -(3/5), z] == -((1048576 ((1 - z)^(4/5) - (1 - z)^(3/5) z^(1/5) + (1 - z)^(2/5) z^(2/5) - (1 - z)^(1/5) z^(3/5) + z^(4/5))^9 (-1171875 + 60546875 z + 1183139908 z^2 + 2750617500 z^3 + 1321739275 z^4 + 102348125 z^5))/ (1171875 ((1 - z)^(1/5) + z^(1/5)) (2 (1 - z)^(2/5) + (-1 + Sqrt[5]) (1 - z)^(1/5) z^(1/5) + 2 z^(2/5))^10 (2 (1 - z)^(2/5) - (1 + Sqrt[5]) (1 - z)^(1/5) z^(1/5) + 2 z^(2/5))^10 (-1 + z)^10)) - (1/(1953125 (1 - z)^(53/5))) (25116 Sqrt[10 - 2 Sqrt[5]] z^(8/5) (6279 + 38640 z + 48300 z^2 + 14000 z^3 + 625 z^4) ArcTan[1 - ((1 - Sqrt[5]) z^(1/5))/ (4 (1 - z)^(1/5)), -((Sqrt[5/8 + Sqrt[5]/8] z^(1/5))/ (1 - z)^(1/5))]) + (1/(1953125 (1 - z)^(53/5))) (25116 Sqrt[2 (5 + Sqrt[5])] z^(8/5) (6279 + 38640 z + 48300 z^2 + 14000 z^3 + 625 z^4) ArcTan[1 - ((1 + Sqrt[5]) z^(1/5))/ (4 (1 - z)^(1/5)), -((Sqrt[5/8 - Sqrt[5]/8] z^(1/5))/ (1 - z)^(1/5))]) + (50232 z^(8/5) (6279 + 38640 z + 48300 z^2 + 14000 z^3 + 625 z^4) Log[1 + z^(1/5)/(1 - z)^(1/5)])/ (1953125 (1 - z)^(53/5)) - (1/(1953125 (1 - z)^(53/5))) (12558 (1 + Sqrt[5]) z^(8/5) (6279 + 38640 z + 48300 z^2 + 14000 z^3 + 625 z^4) Log[1 - ((1 - Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) + z^(2/5)/(1 - z)^(2/5)]) + (1/(1953125 (1 - z)^(53/5))) (12558 (-1 + Sqrt[5]) z^(8/5) (6279 + 38640 z + 48300 z^2 + 14000 z^3 + 625 z^4) Log[1 - ((1 + Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) + z^(2/5)/(1 - z)^(2/5)])








Standard Form





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MathML Form







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5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> <mo> &#8290; </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 10 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 10 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1953125 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 53 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 25116 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 10 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 8 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 625 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14000 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48300 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 38640 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 6279 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mfrac> <mn> 5 </mn> <mn> 8 </mn> </mfrac> <mo> + </mo> <mfrac> <msqrt> <mn> 5 </mn> </msqrt> <mn> 8 </mn> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1953125 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 53 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 25116 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 8 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 625 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14000 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48300 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 38640 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 6279 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <mfrac> <mn> 5 </mn> <mn> 8 </mn> </mfrac> <mo> - </mo> <mfrac> <msqrt> <mn> 5 </mn> </msqrt> <mn> 8 </mn> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 50232 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 8 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 625 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14000 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48300 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 38640 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 6279 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 1953125 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 53 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1953125 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 53 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 12558 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 8 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 625 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14000 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48300 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 38640 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 6279 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1953125 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 53 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 12558 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 8 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 625 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14000 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 48300 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 38640 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 6279 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> </msup> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mroot> <mi> z </mi> <mn> 5 </mn> </mroot> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 5 </mn> </mroot> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 5 </cn> <cn type='integer'> 5 </cn> </list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 5 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1048576 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 102348125 </cn> <apply> <power 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1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 3 <sep /> 5 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 5 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 2 <sep /> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 5 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 5 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 4 <sep /> 5 </cn> </apply> </apply> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1171875 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Rule Form





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





2007-05-02