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 Cos

 http://functions.wolfram.com/01.07.21.2528.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Sin[b z^r + e]^m Cos[c z^r]^v, z] == 2^(-m - v) ((1/\[Alpha]) (z^\[Alpha] Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2])) + (1/r) ((z^\[Alpha] Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[((-1)^k Binomial[m, k] ((E^(2 I e m) Gamma[\[Alpha]/r, I b (2 k - m) z^r])/(I b (2 k - m) z^r)^(\[Alpha]/r) + ((-1)^m E^(4 I e k) Gamma[\[Alpha]/r, I b (-2 k + m) z^r])/ (I b (-2 k + m) z^r)^(\[Alpha]/r)))/E^(I e (2 k + m)), {k, 0, Floor[(1/2) (-1 + m)]}])/I^m) + (1/r) (z^\[Alpha] Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(Binomial[v, s] (((-I) c (2 s - v) z^r)^(\[Alpha]/r) Gamma[\[Alpha]/r, I c (2 s - v) z^r] + (I c (2 s - v) z^r)^(\[Alpha]/r) Gamma[\[Alpha]/r, I c (-2 s + v) z^r]))/(c^2 (-2 s + v)^2 z^(2 r))^(\[Alpha]/r), {s, 0, Floor[(1/2) (-1 + v)]}]) - (1/r) (Sum[(-1)^k Binomial[m, k] Sum[(z^\[Alpha] Binomial[v, s] (((-1)^m E^(4 I e k) Gamma[\[Alpha]/r, (-I) (2 b k - b m + 2 c s - c v) z^r])/((-I) (2 b k - b m + 2 c s - c v) z^r)^ (\[Alpha]/r) + (E^(2 I e m) Gamma[\[Alpha]/r, I (2 b k - b m + 2 c s - c v) z^r])/(I (2 b k - b m + 2 c s - c v) z^r)^ (\[Alpha]/r) + ((-1)^m E^(4 I e k) Gamma[\[Alpha]/r, (-I) (2 b k - b m - 2 c s + c v) z^r])/((-I) (2 b k - b m - 2 c s + c v) z^r)^(\[Alpha]/r) + (E^(2 I e m) Gamma[\[Alpha]/r, I (2 b k - b m - 2 c s + c v) z^r])/(I (2 b k - b m - 2 c s + c v) z^r)^(\[Alpha]/r)))/ E^(I e (2 k + m)), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}]/I^m)) /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z α - 1 sin m ( b z r + e ) cos v ( c z r ) z 2 - m - v ( z α ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) α ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] + 1 r ( - m z α ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) k = 0 m - 1 2 ( - 1 ) k - e ( 2 k + m ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 e m Γ ( α r , b ( 2 k - m ) z r ) ( b ( 2 k - m ) z r ) - α r + ( - 1 ) m 4 e k ( b ( m - 2 k ) z r ) - α r Γ ( α r , b ( m - 2 k ) z r ) ) ) + 1 r ( z α ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) s = 0 v - 1 2 ( c 2 ( v - 2 s ) 2 z 2 r ) - α r ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( α r , c ( 2 s - v ) z r ) ( - c ( 2 s - v ) z r ) α / r + ( c ( 2 s - v ) z r ) α / r Γ ( α r , c ( v - 2 s ) z r ) ) ) - 1 r ( - m k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 - e ( 2 k + m ) z α ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m 4 e k Γ ( α r , - ( 2 b k - b m - 2 c s + c v ) z r ) ( - ( 2 b k - b m - 2 c s + c v ) z r ) - α r + 2 e m ( ( 2 b k - b m - 2 c s + c v ) z r ) - α r Γ ( α r , ( 2 b k - b m - 2 c s + c v ) z r ) + ( - 1 ) m 4 e k ( - ( 2 b k - b m + 2 c s - c v ) z r ) - α r Γ ( α r , - ( 2 b k - b m + 2 c s - c v ) z r ) + 2 e m ( ( 2 b k - b m + 2 c s - c v ) z r ) - α r Γ ( α r , ( 2 b k - b m + 2 c s - c v ) z r ) ) ) ) /; m + v + Condition z z α -1 b z r e m c z r v 2 -1 m -1 v z α \$CellContext`m 2 -1 \$CellContext`v 2 -1 α -1 Binomial m m 2 -1 Binomial v v 2 -1 1 r -1 -1 m z α Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k -1 e 2 k m Binomial m k 2 e m Gamma α r -1 b 2 k -1 m z r b 2 k -1 m z r -1 α r -1 -1 m 4 e k b m -1 2 k z r -1 α r -1 Gamma α r -1 b m -1 2 k z r 1 r -1 z α Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 c 2 v -1 2 s 2 z 2 r -1 α r -1 Binomial v s Gamma α r -1 c 2 s -1 v z r -1 c 2 s -1 v z r α r -1 c 2 s -1 v z r α r -1 Gamma α r -1 c v -1 2 s z r -1 1 r -1 -1 m k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 -1 e 2 k m z α Binomial v s -1 m 4 e k Gamma α r -1 -1 2 b k -1 b m -1 2 c s c v z r -1 2 b k -1 b m -1 2 c s c v z r -1 α r -1 2 e m 2 b k -1 b m -1 2 c s c v z r -1 α r -1 Gamma α r -1 2 b k -1 b m -1 2 c s c v z r -1 m 4 e k -1 2 b k -1 b m 2 c s -1 c v z r -1 α r -1 Gamma α r -1 -1 2 b k -1 b m 2 c s -1 c v z r 2 e m 2 b k -1 b m 2 c s -1 c v z r -1 α r -1 Gamma α r -1 2 b k -1 b m 2 c s -1 c v z r m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18