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; }

 Cos

 http://functions.wolfram.com/01.07.21.2499.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18