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; }

 Log

 http://functions.wolfram.com/01.04.20.0015.01

 Input Form

 D[Log[z]^n, {z, \[Alpha]}] == Piecewise[{{((-1)^(\[Alpha] - 1) n! Sum[(Log[z]^(u - i)/(u - i)!) (Derivative[n - 1 - u][Gamma][1]/(n - 1 - u)!) Sum[Subscript[a, v] Subscript[b, i - v], {v, 0, i}], {u, 0, n - 1}, {i, 0, u}])/z^\[Alpha], Element[\[Alpha], Integers] && \[Alpha] > 0}}, D[(Gamma[1 + a]/Gamma[1 + a - \[Alpha]]) z^(a - \[Alpha]), {a, n}] /; a == 0] /; Element[n, Integers] && n > 0 && Subscript[a, 2 k] == ((-1)^k Pi^(2 k))/(2 k + 1)! && Subscript[a, 2 k + 1] == 0 && Subscript[b, k] == ((-1)^k Derivative[k][Gamma][\[Alpha]])/k! && Element[k, Integers] && k > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "n"]]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], RowBox[List["n", "!"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "u"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["u", "-", "i"]]], RowBox[List[RowBox[List["(", RowBox[List["u", "-", "i"]], ")"]], "!"]]], FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", RowBox[List["n", "-", "1", "-", "u"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "-", "u"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["v", "=", "0"]], "i"], RowBox[List[SubscriptBox["a", "v"], " ", SubscriptBox["b", RowBox[List["i", "-", "v"]]]]]]], ")"]]]]]]]]]], ",", RowBox[List[RowBox[List["Element", "[", RowBox[List["\[Alpha]", ",", "Integers"]], "]"]], "\[And]", RowBox[List["\[Alpha]", ">", "0"]]]]]], "}"]], "}"]], ",", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "n"]], "}"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], "]"]]], SuperscriptBox["z", RowBox[List["a", "-", "\[Alpha]"]]]]], ")"]]]], ")"]], "/;", RowBox[List["a", "\[Equal]", "0"]]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List["2", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[Pi]", RowBox[List["2", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "!"]]]]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "\[Equal]", "0"]], "\[And]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "\[Alpha]", "]"]]]], RowBox[List["k", "!"]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]

 MathML Form

 α log n ( z ) z α ( - 1 ) α - 1 z - α n ! u = 0 n - 1 i = 0 u log u - i ( z ) Γ ( n - u - 1 ) TagBox[RowBox[List["(", RowBox[List["n", "-", "u", "-", "1"]], ")"]], Derivative] ( 1 ) ( u - i ) ! ( n - u - 1 ) ! v = 0 i a v b i - v α + n Γ ( a + 1 ) z a - α Γ ( a - α + 1 ) a n /; a 0 True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; n + a 2 k ( - 1 ) k π 2 k ( 2 k + 1 ) ! a 2 k + 1 0 b k ( - 1 ) k Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( α ) k ! k + Condition z α z n -1 α -1 z -1 α n i 0 u u 0 n -1 z u -1 i D Gamma 1 1 n -1 u -1 u -1 i n -1 u -1 -1 v 0 i Subscript a v Subscript b i -1 v α SuperPlus Condition a n Gamma a 1 z a -1 α Gamma a -1 α 1 -1 a 0 n SuperPlus Subscript a 2 k -1 k 2 k 2 k 1 -1 Subscript a 2 k 1 0 Subscript b k -1 k α k Gamma α k -1 k SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], SuperscriptBox[RowBox[List["Log", "[", "z_", "]"]], "n_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "u"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["u", "-", "i"]]], " ", RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", RowBox[List["n", "-", "1", "-", "u"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "1", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["v", "=", "0"]], "i"], RowBox[List[SubscriptBox["a", "v"], " ", SubscriptBox["b", RowBox[List["i", "-", "v"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["u", "-", "i"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "-", "u"]], ")"]], "!"]]]]]]]]]]], RowBox[List[RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Alpha]", ">", "0"]]]]], List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "n"]], "}"]]]]], FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a"]], "]"]], " ", SuperscriptBox["z", RowBox[List["a", "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], "]"]]]]], "/;", RowBox[List["a", "\[Equal]", "0"]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List[SubscriptBox["a", RowBox[List["2", " ", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "\[Alpha]", "]"]]]], RowBox[List["k", "!"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

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

 2007-05-02