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variants of this functions
Erf




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erf[z] > Representations through more general functions > Through hypergeometric functions > Involving pFq





http://functions.wolfram.com/06.25.26.0018.01









  


  








Input Form





Erf[z] == ((2 z)/Sqrt[Pi]) HypergeometricPFQ[{1/4}, {1/2, 5/4}, z^4/4] - ((2 z^3)/(3 Sqrt[Pi])) HypergeometricPFQ[{3/4}, {3/2, 7/4}, z^4/4]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Erf", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", "z"]], SqrtBox["\[Pi]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["1", "4"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["5", "4"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "4"]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", "3"]]], RowBox[List["3", " ", SqrtBox["\[Pi]"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["3", "4"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["7", "4"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "4"]]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> erf </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;1&quot;], SubscriptBox[&quot;F&quot;, &quot;2&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;5&quot;, &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[SuperscriptBox[&quot;z&quot;, &quot;4&quot;], &quot;4&quot;], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erf", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["1", "4"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["5", "4"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "4"]]], "]"]]]], SqrtBox["\[Pi]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", "3"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox["3", "4"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["7", "4"]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "4"]]], "]"]]]], RowBox[List["3", " ", SqrtBox["\[Pi]"]]]]]]]]]]


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





2007-05-02