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; }

 InverseErfc

 http://functions.wolfram.com/06.31.20.0003.01

 Input Form

 D[InverseErfc[z], {z, n}] == KroneckerDelta[n] InverseErfc[z] - (Pi^(n/2)/2^n) E^(n InverseErfc[z]^2) Sum[\[Ellipsis] Sum[KroneckerDelta[Sum[(i - 1) Subscript[j, i], {i, 2, n}], n - 1] (-1)^Sum[Subscript[j, i], {i, 2, n}] (n - 1 + Sum[Subscript[j, i], {i, 2, n}])! Product[(1/Subscript[j, i]!) (((-1)^(i - 1) 2^(-1 + i) E^InverseErfc[z]^2 Sqrt[Pi] InverseErfc[z]^(1 - i))/i!)^ Subscript[j, i] HypergeometricPFQRegularized[{1/2, 1}, {1 - i/2, (3 - i)/2}, -InverseErfc[z]^2]^Subscript[j, i], {i, 2, n}], {Subscript[j, n], 0, n}], {Subscript[j, 2], 0, n}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["InverseErfc", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["InverseErfc", "[", "z", "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", RowBox[List["n", "/", "2"]]], SuperscriptBox["2", "n"]], SuperscriptBox["\[ExponentialE]", RowBox[List["n", " ", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], "n"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "n"], "=", "0"]], "n"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], RowBox[List[RowBox[List["(", RowBox[List["i", "-", "1"]], ")"]], " ", SubscriptBox["j", "i"]]]]], ",", RowBox[List["n", "-", "1"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], SubscriptBox["j", "i"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], SubscriptBox["j", "i"]]]]], ")"]], "!"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "2"]], "n"], RowBox[List[FractionBox["1", RowBox[List[SubscriptBox["j", "i"], "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "-", "1"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "i"]]], " ", SuperscriptBox["\[ExponentialE]", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], "2"]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], RowBox[List["1", "-", "i"]]]]], RowBox[List["i", "!"]]], ")"]], SubscriptBox["j", "i"]], SuperscriptBox[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["i", "2"]]], ",", FractionBox[RowBox[List["3", "-", "i"]], "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], "2"]]]]], "]"]], SubscriptBox["j", "i"]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n erfc - 1 ( z ) z n erfc - 1 ( z ) δ KroneckerDelta n - π n / 2 2 n n erfc - 1 ( z ) 2 j 2 = 0 n j n = 0 n δ KroneckerDelta i = 2 n ( i - 1 ) j i , n - 1 ( - 1 ) i = 2 n j i ( n + i = 2 n j i - 1 ) ! i = 2 n 1 j i ! ( ( - 1 ) i - 1 2 i - 1 erfc - 1 ( z ) 2 π erfc - 1 ( z ) 1 - i i ! ) j i 2 F ~ 2 ( 1 2 , 1 ; 1 - i 2 , 3 - i 2 ; - erfc - 1 ( z ) 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["i", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "i"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox[RowBox[List[SuperscriptBox["erfc", RowBox[List["-", "1"]]], "(", "z", ")"]], "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] j i /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition z n InverseErfc z InverseErfc z KroneckerDelta n -1 n 2 -1 2 n -1 n InverseErfc z 2 Subscript j 2 0 n Subscript j n 0 n KroneckerDelta i 2 n i -1 Subscript j i n -1 -1 i 2 n Subscript j i n i 2 n Subscript j i -1 i 2 n 1 Subscript j i -1 -1 i -1 2 i -1 InverseErfc z 2 1 2 InverseErfc z 1 -1 i i -1 Subscript j i HypergeometricPFQRegularized 1 2 1 1 -1 i 2 -1 3 -1 i 2 -1 -1 InverseErfc z 2 Subscript j i n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["InverseErfc", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["InverseErfc", "[", "z", "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["n", "/", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["n", " ", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], "n"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "n"], "=", "0"]], "n"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], RowBox[List[RowBox[List["(", RowBox[List["i", "-", "1"]], ")"]], " ", SubscriptBox["j", "i"]]]]], ",", RowBox[List["n", "-", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], SubscriptBox["j", "i"]]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], SubscriptBox["j", "i"]]]]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "2"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "i"]]], " ", SuperscriptBox["\[ExponentialE]", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], "2"]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], RowBox[List["1", "-", "i"]]]]], RowBox[List["i", "!"]]], ")"]], SubscriptBox["j", "i"]], " ", SuperscriptBox[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["i", "2"]]], ",", FractionBox[RowBox[List["3", "-", "i"]], "2"]]], "}"]], ",", RowBox[List["-", SuperscriptBox[RowBox[List["InverseErfc", "[", "z", "]"]], "2"]]]]], "]"]], SubscriptBox["j", "i"]]]], RowBox[List[SubscriptBox["j", "i"], "!"]]]]]]]]]]]]]]], SuperscriptBox["2", "n"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02