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variants of this functions
Hypergeometric2F1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1[a,b,c,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Generic formulas for main term





http://functions.wolfram.com/07.23.06.0064.01









  


  










Input Form





Hypergeometric2F1[a, b, c, z] \[Proportional] Piecewise[{{((-1)^(a - c) (b - c)! Gamma[c])/((-z)^b (Gamma[a] (b - a)!)), Element[b - a, Integers] && b - a >= 0 && Element[a - c, Integers] && a - c >= 0}, {(Gamma[c] Log[-z])/((-z)^a (Gamma[a] (c - a - 1)!)), b == a && !(Element[a - c, Integers] && a - c >= 0)}, {((-1)^(b - c) (a - c)! Gamma[c])/((-z)^a (Gamma[b] (a - b)!)), Element[a - b, Integers] && a - b >= 0 && Element[b - c, Integers] && b - c >= 0}, {(Gamma[b - a] Gamma[c])/((-z)^a (Gamma[b] Gamma[c - a])), Re[b - a] > 0 || (Element[-c, Integers] && -c >= 0 && Element[-a, Integers] && -a >= 0 && c - a <= 0)}, {(Gamma[a - b] Gamma[c])/((-z)^b (Gamma[a] Gamma[c - b])), Re[b - a] > 0 || (Element[-c, Integers] && -c >= 0 && Element[-b, Integers] && -b >= 0 && c - b <= 0)}, {ComplexInfinity, Element[-c, Integers] && -c >= 0 && ((Element[-a, Integers] && -a >= 0 && c - a > 0) || (Element[-b, Integers] && -b >= 0 && c - b > 0))}}, (Gamma[b - a] Gamma[c])/((-z)^a (Gamma[b] Gamma[c - a])) + (Gamma[a - b] Gamma[c])/((-z)^b (Gamma[a] Gamma[c - b]))] /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Hypergeometric2F1 </ci> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <piecewise> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <ci> &#8469; </ci> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> b </ci> <ci> a </ci> </apply> <apply> <notin /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <ci> &#8469; </ci> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <ci> &#8469; </ci> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <or /> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <geq /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </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> <or /> <apply> <gt /> <apply> <real /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <geq /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </piece> <piece> <apply> <ci> OverTilde </ci> <infinity /> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <or /> <apply> <and /> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <gt /> <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> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <gt /> <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> <otherwise> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </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> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </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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