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variants of this functions
InverseWeierstrassP






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseWeierstrassP[z,{g2,g3}] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/09.22.06.0004.01









  


  










Input Form





InverseWeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] == InverseWeierstrassP[Subscript[z, 0], {Subscript[g, 2], Subscript[g, 3]}] + Sum[(1/k!) (KroneckerDelta[k - 1]/Sqrt[4 Subscript[z, 0]^3 - Subscript[g, 2] Subscript[z, 0] - Subscript[g, 3]] + Sum[(1/m!) Pochhammer[1/2 - m, m] Sum[(-1)^j Binomial[m, j] (4 Subscript[z, 0]^3 - Subscript[g, 2] Subscript[z, 0] - Subscript[g, 3])^(j - m - 1/2) Sum[(-1)^(k + Subscript[k, 2] + Subscript[k, 3] - 1) KroneckerDelta[m - j, Subscript[k, 1] + Subscript[k, 2] + Subscript[k, 3]] Multinomial[Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]] 4^Subscript[k, 1] Subscript[g, 2]^ Subscript[k, 2] Subscript[g, 3]^Subscript[k, 3] Pochhammer[-3 Subscript[k, 1] - Subscript[k, 2], k - 1] z^(1 + 3 Subscript[k, 1] + Subscript[k, 2] - k), {Subscript[k, 1], 0, m - j}, {Subscript[k, 2], 0, m - j}, {Subscript[k, 3], 0, m - j}], {j, 0, m - 1}], {m, 1, k - 1}]) (z - Subscript[z, 0])^k, {k, 1, Infinity}]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> &#8472; </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <msup> <mi> &#8472; </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> InverseWeierstrassP </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <times /> <apply> <ci> InverseWeierstrassP </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseWeierstrassP", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["InverseWeierstrassP", "[", RowBox[List[SubscriptBox["zz", "0"], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List["k", "-", "1"]], "]"]], SqrtBox[RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["zz", "0", "3"]]], "-", RowBox[List[SubscriptBox["gg", "2"], " ", SubscriptBox["zz", "0"]]], "-", SubscriptBox["gg", "3"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "m"]], ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["zz", "0", "3"]]], "-", RowBox[List[SubscriptBox["gg", "2"], " ", SubscriptBox["zz", "0"]]], "-", SubscriptBox["gg", "3"]]], ")"]], RowBox[List["j", "-", "m", "-", FractionBox["1", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], RowBox[List["m", "-", "j"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], RowBox[List["m", "-", "j"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "3"], "=", "0"]], RowBox[List["m", "-", "j"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"], "-", "1"]]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["m", "-", "j"]], ",", RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]]]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", "1"], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], "]"]], " ", SuperscriptBox["4", SubscriptBox["k", "1"]], " ", SubsuperscriptBox["gg", "2", SubscriptBox["k", "2"]], " ", SubsuperscriptBox["gg", "3", SubscriptBox["k", "3"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubscriptBox["k", "1"]]], "-", SubscriptBox["k", "2"]]], ",", RowBox[List["k", "-", "1"]]]], "]"]], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["3", " ", SubscriptBox["k", "1"]]], "+", SubscriptBox["k", "2"], "-", "k"]]]]]]]]]]]]]]]]], RowBox[List["m", "!"]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]]]]










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





2007-05-02





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