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 | | http://functions.wolfram.com/07.23.09.0001.01 | 
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 | | Hypergeometric2F1[a, b, c, z] == Limit[HypergeometricPFQ[{a, b, p z}, {c, p}, 
    1], p -> Infinity] /; Re[c - a - b + p (1 - z)] > 0 | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", RowBox[List["p", " ", "z"]]]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "p"]], "}"]], ",", "1"]], "]"]], ",", RowBox[List["p", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["Re", "[", RowBox[List["c", "-", "a", "-", "b", "+", RowBox[List["p", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]]]], "]"]], ">", "0"]]]]]] | 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 2 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 1 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> b </mi>  </mrow>  <mo> ; </mo>  <mi> c </mi>  <mo> ; </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <munder>  <mi> lim </mi>  <mrow>  <mi> p </mi>  <semantics>  <mo> → </mo>  <annotation encoding='Mathematica'> "\[Rule]" </annotation>  </semantics>  <mi> ∞ </mi>  </mrow>  </munder>  <mo> ⁢ </mo>  <mtext>   </mtext>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 3 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> p </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  </mrow>  <mo> ; </mo>  <mrow>  <mi> c </mi>  <mo> , </mo>  <mi> p </mi>  </mrow>  <mo> ; </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["p", " ", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["c", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["p", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation>  </semantics>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <mi> Re </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> + </mo>  <mrow>  <mi> p </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> > </mo>  <mn> 0 </mn>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <ci> Hypergeometric2F1 </ci>  <ci> a </ci>  <ci> b </ci>  <ci> c </ci>  <ci> z </ci>  </apply>  <apply>  <limit />  <bvar>  <ci> p </ci>  </bvar>  <condition>  <apply>  <tendsto />  <ci> p </ci>  <infinity />  </apply>  </condition>  <apply>  <ci> HypergeometricPFQ </ci>  <list>  <ci> a </ci>  <ci> b </ci>  <apply>  <times />  <ci> p </ci>  <ci> z </ci>  </apply>  </list>  <list>  <ci> c </ci>  <ci> p </ci>  </list>  <cn type='integer'> 1 </cn>  </apply>  </apply>  </apply>  <apply>  <gt />  <apply>  <real />  <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>  <times />  <ci> p </ci>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  <cn type='integer'> 0 </cn>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", RowBox[List["p", " ", "z"]]]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", "p"]], "}"]], ",", "1"]], "]"]], ",", RowBox[List["p", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List["Re", "[", RowBox[List["c", "-", "a", "-", "b", "+", RowBox[List["p", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]]]], "]"]], ">", "0"]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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 | | HypergeometricPFQ[{},{},z] |  | HypergeometricPFQ[{},{b},z] |  | HypergeometricPFQ[{a},{},z] |  | HypergeometricPFQ[{a},{b},z] |  | HypergeometricPFQ[{a1},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] |  | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] |  | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] |  | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] |  | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] |  |  | 
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