Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











AppellF1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > AppellF1[a,b1,b2,c,z1,z2] > General characteristics > Branch points > With respect to zk





http://functions.wolfram.com/07.36.04.0015.01









  


  










Input Form





RamificationIndex[AppellF1[a, Subscript[b, 1], Subscript[b, 2], c, Subscript[z, 1], Subscript[z, 2]], Subscript[z, k], ComplexInfinity] == LCM[s, u] /; a == r/s && Subscript[b, k] == t/u && Element[{r, s, t, u}, Integers] && s > 1 && u > 1 && GCD[r, s] == 1 && GCD[t, u] == 1 && NonTerminatingHypergeometricSeriesQ[ {a, Subscript[b, k]}] && Element[k, {1, 2}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "c", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], ",", SubscriptBox["z", "k"], ",", "ComplexInfinity"]], "]"]], "\[Equal]", RowBox[List["LCM", "[", RowBox[List["s", ",", "u"]], "]"]]]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", FractionBox["r", "s"]]], "\[And]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox["t", "u"]]], "\[And]", RowBox[List["Element", "[", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s", ",", "t", ",", "u"]], "}"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List["s", ">", "1"]], "\[And]", RowBox[List["u", ">", "1"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["r", ",", "s"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["t", ",", "u"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["NonTerminatingHypergeometricSeriesQ", "[", RowBox[List["{", RowBox[List["a", ",", SubscriptBox["b", "k"]]], "}"]], "]"]], "\[And]", RowBox[List["k", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2"]], "}"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> &#8475; </mi> <msub> <mi> z </mi> <mi> k </mi> </msub> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mover> <mi> &#8734; </mi> <mo> ~ </mo> </mover> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> lcm </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#10869; </mo> <mfrac> <mi> r </mi> <mi> s </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mfrac> <mi> t </mi> <mi> u </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> t </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> u </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#119977;&#119983; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> { </mo> <mrow> <mi> a </mi> <mo> , </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> } </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <mi> &#8475; </mi> <msub> <mi> z </mi> <mi> k </mi> </msub> </msub> <mo> ( </mo> <mrow> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mover> <mi> &#8734; </mi> <mo> ~ </mo> </mover> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> lcm </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#10869; </mo> <mfrac> <mi> r </mi> <mi> s </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> &#10869; </mo> <mfrac> <mi> t </mi> <mi> u </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> t </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> u </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8800; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> , </mo> <mi> u </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#119977;&#119983; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> { </mo> <mrow> <mi> a </mi> <mo> , </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> } </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mrow> <mo> { </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> } </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a_", ",", SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", "c_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], ",", SubscriptBox["z_", "k"], ",", "ComplexInfinity"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["LCM", "[", RowBox[List["s", ",", "u"]], "]"]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", FractionBox["r", "s"]]], "&&", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox["t", "u"]]], "&&", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s", ",", "t", ",", "u"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["s", ">", "1"]], "&&", RowBox[List["u", ">", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["r", ",", "s"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["t", ",", "u"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["NonTerminatingHypergeometricSeriesQ", "[", RowBox[List["{", RowBox[List["a", ",", SubscriptBox["b", "k"]]], "}"]], "]"]], "&&", RowBox[List["k", "\[Element]", RowBox[List["{", RowBox[List["1", ",", "2"]], "}"]]]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.