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 Sin

 http://functions.wolfram.com/01.06.20.0015.01

 Input Form

 D[(z^\[Beta] Sin[z])^a, {z, \[Alpha]}] == E^(2 I a Pi Floor[1/2 - Arg[Sin[z]/z]/(2 Pi) - Im[(\[Beta] + 1) Log[z]]/ (2 Pi)]) a z^((\[Beta] + 1) a - \[Alpha]) Sum[Binomial[k - a, k] Sum[(((-1)^j Binomial[k, j])/(a - j)) Subscript[p, j, k] FDPowerConstant[z, (\[Beta] + 1) a + k, \[Alpha]] z^k, {j, 0, k}], {k, 0, Infinity}] /; Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j m - k + m) (1 + 2 Floor[m/2] - m) ((-1)^(m/2)/(m + 1)!) Subscript[p, j, k - m], {m, 1, 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["(", RowBox[List[SuperscriptBox["z", "\[Beta]"], RowBox[List["Sin", "[", "z", "]"]]]], ")"]], "a"]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["Sin", "[", "z", "]"]], "z"], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Im", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], "a", " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], " ", "a"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "a"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], RowBox[List["a", "-", "j"]]], SubscriptBox["p", RowBox[List["j", ",", "k"]]], RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], " ", "a"]], "+", "k"]], ",", "\[Alpha]"]], "]"]], SuperscriptBox["z", "k"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", RowBox[List["Floor", "[", FractionBox["m", "2"], "]"]]]], "-", "m"]], ")"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "/", "2"]]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]

 MathML Form

 α ( z β sin ( z ) ) a z α 2 a π 1 2 - arg ( sin ( z ) z ) 2 π - Im ( ( β + 1 ) log ( z ) ) 2 π a z ( β + 1 ) a - α k = 0 ( k - a k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", "a"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j = 0 k ( - 1 ) j a - j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p j , k ℱ𝒞 exp ( α ) ( z , ( β + 1 ) a + k ) z k /; p j , 0 1 p j , k 1 k m = 1 k ( j m + m - k ) ( - m + 2 m 2 + 1 ) ( - 1 ) m / 2 p j , k - m ( m + 1 ) ! k + Condition z α z β z a 2 a 1 2 -1 z z -1 2 -1 -1 β 1 z 2 -1 a z β 1 a -1 α k 0 Binomial k -1 a k j 0 k -1 j a -1 j -1 Binomial k j Subscript p j k Subscript ℱ𝒞 exp α z β 1 a k z k Subscript p j 0 1 Subscript p j k 1 k -1 m 1 k j m m -1 k -1 m 2 m 2 -1 1 -1 m 2 -1 Subscript p j k -1 m m 1 -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["(", RowBox[List[SuperscriptBox["z_", "\[Beta]_"], " ", RowBox[List["Sin", "[", "z_", "]"]]]], ")"]], "a_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["Sin", "[", "z", "]"]], "z"], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Im", "[", RowBox[List[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], " ", "a", " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], " ", "a"]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "a"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]], " ", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], " ", "a"]], "+", "k"]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List["a", "-", "j"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["m", "2"], "]"]]]], "-", "m"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "/", "2"]]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]], "k"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

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

 2007-05-02