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variants of this functions
Hypergeometric2F1Regularized






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1Regularized[a,b,c,z] > Series representations > Generalized power series > Expansions at z==1 > For the function itself > Generic formulas for main term





http://functions.wolfram.com/07.24.06.0062.01









  


  










Input Form





Hypergeometric2F1Regularized[a, b, c, z] \[Proportional] Piecewise[{{0, Element[-c, Integers] && -c >= 0 && ((Element[-a, Integers] && -a >= 0 && c - a <= 0) && (Element[-b, Integers] && -b >= 0 && c - b <= 0))}, {((a + b - c - 1)!/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b) + ((-1)^(a + b - c - 1)/((a + b - c)! Gamma[c - a] Gamma[c - b])) (Log[1 - z] + EulerGamma + PolyGamma[a] + PolyGamma[b] - PolyGamma[1 + a + b - c]), Element[a + b - c, Integers] && a + b - c > 0}, {(-(1/(Gamma[a] Gamma[b]))) (Log[1 - z] + 2 EulerGamma + PolyGamma[a] + PolyGamma[b]), c == a + b}, {(c - a - b - 1)!/(Gamma[c - a] Gamma[c - b]) - (1/((c - a - b)! Gamma[a] Gamma[b])) (z - 1)^(c - a - b) (Log[1 - z] + EulerGamma - PolyGamma[1 + c - a - b] + PolyGamma[c - a] + PolyGamma[c - b]), Element[c - a - b, Integers] && c - a - b > 0}}, Gamma[c - a - b]/(Gamma[c - a] Gamma[c - b]) + (Gamma[a + b - c]/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b)] /; (z -> 1)










Standard Form





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MathML Form







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</ci> </apply> <apply> <and /> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <leq /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <and /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#8469; </ci> </apply> <apply> <leq /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </piece> <piece> <apply> <plus /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </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> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> a </ci> </apply> <apply> <ci> PolyGamma </ci> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <eulergamma /> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; 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</ci> </apply> </apply> </piece> <otherwise> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





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