Sine: Integration (formula 01.06.21.1124)

Sin

$Mathematica Notation$

Elementary Functions

Sin[z]

Integration

Indefinite integration

Involving functions of the direct function and a power function

Involving products of the direct function and a power function

Involving products of several direct functions and a power function

Involving z^alpha-1 ∏ _k=1ⁿsin(a_k z)

http://functions.wolfram.com/01.06.21.1124.01

Input Form

Integrate[z^(Î± - 1)*Product[Sin[Subscript[a, k]*z], {k, 1, n}], z] == (-2)^(-n - 1)*z^Î±*Sum[(E^((I/2)*Pi*Sum[Subscript[k, j], {j, 1, n}])* Gamma[Î±, (-I)*z*Sum[Subscript[k, j]* Subscript[a, j], {j, 1, n}]])/ ((-I)*a*Sum[Subscript[k, j]*Subscript[a, j], {j, 1, n}])^Î± + Gamma[Î±, I*z*Sum[Subscript[k, j]*Subscript[a, j], {j, 1, n}]]/ (E^((I/2)*Pi*Sum[Subscript[k, j], {j, 1, n}])* (I*z*Sum[Subscript[k, j]*Subscript[a, j], {j, 1, n}])^ Î±), {Subscript[k, 1], -1, 1, 2}, {Subscript[k, 2], -1, 1, 2}, â€¦, {Subscript[k, n], -1, 1, 2}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[RowBox[List["Sin", "[", RowBox[List[SubscriptBox["a", "k"], " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], RowBox[List[RowBox[List["-", "n"]], "-", "1"]]], SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", UnderscriptBox[RowBox[List[SubscriptBox["k", "1"], "=", RowBox[List["-", "1"]]]], RowBox[List[RowBox[List["\[CapitalDelta]", " ", SubscriptBox["k", "1"]]], "=", "2"]]], "1"], RowBox[List[UnderoverscriptBox["\[Sum]", UnderscriptBox[RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List["-", "1"]]]], RowBox[List[RowBox[List["\[CapitalDelta]", " ", SubscriptBox["k", "2"]]], "=", "2"]]], "1"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", UnderscriptBox[RowBox[List[SubscriptBox["k", "n"], "=", RowBox[List["-", "1"]]]], RowBox[List[RowBox[List["\[CapitalDelta]", " ", SubscriptBox["k", "n"]]], "=", "2"]]], "1"], " ", RowBox[List["(", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SubscriptBox["k", "j"], SubscriptBox["a", "j"]]]]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], SubscriptBox["k", "j"]]]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[ImaginaryI]", " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SubscriptBox["k", "j"], SubscriptBox["a", "j"]]]]]]]]], "]"]]]], " ", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SubscriptBox["k", "j"], SubscriptBox["a", "j"]]]]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], RowBox[List["Exp", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], SubscriptBox["k", "j"]]]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SubscriptBox["k", "j"], SubscriptBox["a", "j"]]]]]]]]], "]"]]]]]], ")"]], ")"]]]]]]]]]]]]]]]]

MathML Form

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

2002-12-18