html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 InverseWeierstrassP

 http://functions.wolfram.com/09.23.20.0004.01

 Input Form

 D[InverseWeierstrassP[{Subscript[z, 1], Subscript[z, 2]}, {Subscript[g, 2], Subscript[g, 3]}], {Subscript[z, 1], n}] == InverseWeierstrassP[{Subscript[z, 1], Subscript[z, 2]}, {Subscript[g, 2], Subscript[g, 3]}] KroneckerDelta[n] + KroneckerDelta[n - 1]/Sqrt[4 Subscript[z, 1]^3 - Subscript[g, 2] Subscript[z, 1] - Subscript[g, 3]] + Sum[(1/m!) Pochhammer[1/2 - m, m] Sum[(-1)^j Binomial[m, j] (4 Subscript[z, 1]^3 - Subscript[g, 2] Subscript[z, 1] - Subscript[g, 3])^(j - m - 1/2) Sum[(-1)^(n + 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], n - 1] Subscript[z, 1]^(1 + 3 Subscript[k, 1] + Subscript[k, 2] - n), {Subscript[k, 1], 0, m - j}, {Subscript[k, 2], 0, m - j}, {Subscript[k, 3], 0, m - j}], {j, 0, m - 1}], {m, 1, n - 1}] /; Element[n, Integers] && n >= 0 && Subscript[z, 2] == Sqrt[4 Subscript[z, 1]^3 - Subscript[g, 2] Subscript[z, 1] - Subscript[g, 3]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["z", "1"], ",", "n"]], "}"]]], RowBox[List["InverseWeierstrassP", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["InverseWeierstrassP", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]], SqrtBox[RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["z", "1", "3"]]], "-", RowBox[List[SubscriptBox["g", "2"], " ", SubscriptBox["z", "1"]]], "-", SubscriptBox["g", "3"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["m", "!"]]], 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["z", "1", "3"]]], "-", RowBox[List[SubscriptBox["g", "2"], " ", SubscriptBox["z", "1"]]], "-", SubscriptBox["g", "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["n", "+", 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["g", "2", SubscriptBox["k", "2"]], " ", SubsuperscriptBox["g", "3", SubscriptBox["k", "3"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubscriptBox["k", "1"]]], "-", SubscriptBox["k", "2"]]], ",", RowBox[List["n", "-", "1"]]]], "]"]], " ", SubsuperscriptBox["z", "1", RowBox[List["1", "+", RowBox[List["3", " ", SubscriptBox["k", "1"]]], "+", SubscriptBox["k", "2"], "-", "n"]]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["z", "2"], "\[Equal]", SqrtBox[RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["z", "1", "3"]]], "-", RowBox[List[SubscriptBox["g", "2"], " ", SubscriptBox["z", "1"]]], "-", SubscriptBox["g", "3"]]]]]]]]]]]]

 MathML Form

 n - 1 ( z 1 , z 2 ; g 2 , g 3 ) z 1 n δ KroneckerDelta n - 1 4 z 1 3 - g 2 z 1 - g 3 + - 1 ( z 1 , z 2 ; g 2 , g 3 ) δ KroneckerDelta n + m = 1 n - 1 1 m ! ( 1 2 - m ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "m"]], ")"]], "m"], Pochhammer] j = 0 m - 1 ( - 1 ) j ( m j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 4 z 1 3 - g 2 z 1 - g 3 ) j - m - 1 2 k 1 = 0 m - j k 2 = 0 m - j k 3 = 0 m - j ( - 1 ) n + k 2 + k 3 - 1 δ KroneckerDelta m - j , k 1 + k 2 + k 3 ( k 1 + k 2 + k 3 ; k 1 , k 2 , k 3 ) TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["k", "2"], "+", SubscriptBox["k", "3"]]], ";", SubscriptBox["k", "1"]]], ",", SubscriptBox["k", "2"], ",", SubscriptBox["k", "3"]]], ")"]], Multinomial, Rule[Editable, False]] 4 k 1 g 2 k 2 g 3 k 3 ( - 3 k 1 - k 2 ) n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SubscriptBox["k", "1"]]], "-", SubscriptBox["k", "2"]]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] z 1 - n + 3 k 1 + k 2 + 1 /; n z 2 4 z 1 3 - g 2 z 1 - g 3 Condition D InverseWeierstrassP Subscript z 1 Subscript z 2 Subscript g 2 Subscript g 3 Subscript z 1 n KroneckerDelta n -1 4 Subscript z 1 3 -1 Subscript g 2 Subscript z 1 -1 Subscript g 3 1 2 -1 InverseWeierstrassP Subscript z 1 Subscript z 2 Subscript g 2 Subscript g 3 KroneckerDelta n m 1 n -1 1 m -1 Pochhammer 1 2 -1 m m j 0 m -1 -1 j Binomial m j 4 Subscript z 1 3 -1 Subscript g 2 Subscript z 1 -1 Subscript g 3 j -1 m -1 1 2 Subscript k 3 0 m -1 j Subscript k 2 0 m -1 j Subscript k 1 0 m -1 j -1 n Subscript k 2 Subscript k 3 -1 KroneckerDelta m -1 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 -1 Subscript k 2 n -1 Subscript z 1 -1 n 3 Subscript k 1 Subscript k 2 1 n Subscript z 2 4 Subscript z 1 3 -1 Subscript g 2 Subscript z 1 -1 Subscript g 3 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z_", "1"], ",", "n_"]], "}"]]]]], RowBox[List["InverseWeierstrassP", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["InverseWeierstrassP", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["gg", "2"], ",", SubscriptBox["gg", "3"]]], "}"]]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", FractionBox[RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]], SqrtBox[RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["zz", "1", "3"]]], "-", RowBox[List[SubscriptBox["gg", "2"], " ", SubscriptBox["zz", "1"]]], "-", SubscriptBox["gg", "3"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], RowBox[List["n", "-", "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", "1", "3"]]], "-", RowBox[List[SubscriptBox["gg", "2"], " ", SubscriptBox["zz", "1"]]], "-", 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["n", "+", 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["n", "-", "1"]]]], "]"]], " ", SubsuperscriptBox["zz", "1", RowBox[List["1", "+", RowBox[List["3", " ", SubscriptBox["k", "1"]]], "+", SubscriptBox["k", "2"], "-", "n"]]]]]]]]]]]]]]]]], RowBox[List["m", "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[SubscriptBox["zz", "2"], "\[Equal]", SqrtBox[RowBox[List[RowBox[List["4", " ", SubsuperscriptBox["zz", "1", "3"]]], "-", RowBox[List[SubscriptBox["gg", "2"], " ", SubscriptBox["zz", "1"]]], "-", SubscriptBox["gg", "3"]]]]]]]]]]]]]]

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

 2007-05-02