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/01.04.20.0022.01

 Input Form

 D[z^a Log[z]^n, {z, \[Alpha]}] == Piecewise[{{(-1)^(\[Alpha] - a - 1) z^(a - \[Alpha]) n! Sum[(Log[z]^(u - i)/(u - i)!) (Derivative[n - 1 - u][Gamma][a + 1]/ (n - 1 - u)!) Sum[Subscript[a, v] Subscript[b, i - v], {v, 0, i}], {u, 0, n - 1}, {i, 0, u}], Element[\[Alpha] - a, Integers] && \[Alpha] - a > 0}, {((Pi (-1)^(a - 1) n! z^(a - \[Alpha]))/ ((-a - 1)! Gamma[1 + a - \[Alpha]])) Sum[(Log[z]^(n + 1 - u)/(n + 1 - u)!) (u - i + 1) Sum[(v + 1) Sum[((-1)^(r + v) Binomial[v, r] Subscript[p, r, v] Subscript[c, i - v - 1])/(1 + r), {r, 0, v}], {v, 0, i}] Sum[((-1)^s Binomial[u - i, s] Subscript[q, s, u - i])/(1 + s), {s, 0, u - i}], {u, 0, n + 1}, {i, 0, u}], Element[-a, Integers] && -a > 0}}, D[(Gamma[1 + a]/Gamma[1 + a - \[Alpha]]) z^(a - \[Alpha]), {a, n}]] /; 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] - a])/k! && Element[k, Integers] && k > 0 && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j i - k + i) Subscript[e, i] Subscript[p, j, k - i], {i, 1, k}] && Subscript[e, k] == Derivative[k][Gamma][-a]/((-a - 1)! k!) && Subscript[q, j, 0] == 1 && Subscript[q, j, k] == (1/k) Sum[(j i - k + i) Subscript[d, i] Subscript[q, j, k - i], {i, 1, k}] && Subscript[d, k] == Derivative[k][Gamma][1 + a - \[Alpha]]/(Gamma[1 + a - \[Alpha]] k!) && Subscript[c, 2 k + 1] == ((-1)^k 2 (2^(2 k + 1) - 1) BernoulliB[2 k + 2] Pi^(2 k + 1))/(2 k + 2)! && Subscript[c, 2 k] == 0 && Element[k, Integers] && k > 0

 Standard Form

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 MathML Form

 α ( z a log n ( z ) ) z α ( - 1 ) - a + α - 1 z a - 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] ( a + 1 ) v = 0 i a v b i - v ( u - i ) ! ( n - u - 1 ) ! α - a + π ( - 1 ) a - 1 n ! z a - α ( - a - 1 ) ! Γ ( a - α + 1 ) u = 0 n + 1 i = 0 u log n - u + 1 ( z ) ( - i + u + 1 ) ( n - u + 1 ) ! ( v = 0 i ( v + 1 ) r = 0 v ( - 1 ) r + v ( v r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["r", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p r , v c i - v - 1 r + 1 ) s = 0 u - i ( - 1 ) s ( u - i s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["u", "-", "i"]], Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] q s , u - i s + 1 - a + n Γ ( a + 1 ) z a - α Γ ( a - α + 1 ) a n 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 - a ) k ! k + p j , 0 1 p j , k 1 k i = 1 k ( j i + i - k ) e i p j , k - i e k Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( - a ) ( - a - 1 ) ! k ! q j , 0 1 q j , k 1 k i = 1 k ( j i + i - k ) d i q j , k - i d k Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( a - α + 1 ) Γ ( a - α + 1 ) k ! c 2 k + 1 ( - 1 ) k 2 ( 2 2 k + 1 - 1 ) B TagBox["B", BernoulliB] 2 k + 2 π 2 k + 1 ( 2 k + 2 ) ! c 2 k 0 k + Condition z α z a z n -1 -1 a α -1 z a -1 α n i 0 u u 0 n -1 z u -1 i D Gamma a 1 a 1 n -1 u -1 v 0 i Subscript a v Subscript b i -1 v u -1 i n -1 u -1 -1 α -1 a SuperPlus -1 a -1 n z a -1 α -1 a -1 Gamma a -1 α 1 -1 i 0 u u 0 n 1 z n -1 u 1 -1 i u 1 n -1 u 1 -1 v 0 i v 1 r 0 v -1 r v Binomial v r Subscript p r v Subscript c i -1 v -1 r 1 -1 s 0 u -1 i -1 s Binomial u -1 i s Subscript q s u -1 i s 1 -1 -1 a SuperPlus a n Gamma a 1 z a -1 α Gamma a -1 α 1 -1 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 D Gamma α -1 a α -1 a k k -1 k SuperPlus Subscript p j 0 1 Subscript p j k 1 k -1 i 1 k j i i -1 k Subscript e i Subscript p j k -1 i Subscript e k D Gamma -1 a -1 a k -1 a -1 k -1 Subscript q j 0 1 Subscript q j k 1 k -1 i 1 k j i i -1 k Subscript d i Subscript q j k -1 i Subscript d k D Gamma a -1 α 1 a -1 α 1 k Gamma a -1 α 1 k -1 Subscript c 2 k 1 -1 k 2 2 2 k 1 -1 BernoulliB 2 k 2 2 k 1 2 k 2 -1 Subscript c 2 k 0 k SuperPlus [/itex]

 Rule Form

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

 2007-05-02