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 Cosh

 http://functions.wolfram.com/01.20.21.3649.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Cos[d + c z]^m Cosh[a z]^v, z] == 2^(-m - v) (z^\[Alpha]/\[Alpha]) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) - 2^(-m - v) z^\[Alpha] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[E^(-2 I d k - I d m) Binomial[m, k] ((E^(2 I d m) Gamma[\[Alpha], (-I) c (-2 k + m) z])/ ((-I) c (-2 k + m) z)^\[Alpha] + (E^(4 I d k) Gamma[\[Alpha], (-2 I c k + I c m) z])/ ((-2 I c k + I c m) z)^\[Alpha]), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(-m - v) z^\[Alpha] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (Gamma[\[Alpha], (-a) (-2 s + v) z]/ ((-a) (-2 s + v) z)^\[Alpha] + Gamma[\[Alpha], (-2 a s + a v) z]/ ((-2 a s + a v) z)^\[Alpha]), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-m - v) z^\[Alpha] Sum[Binomial[m, k] Sum[E^(-2 I d k - I d m) Binomial[v, s] (((-E^(2 I d m)) Gamma[\[Alpha], (2 I c k - I c m + 2 a s - a v) z])/ ((2 I c k - I c m + 2 a s - a v) z)^\[Alpha] - (E^(4 I d k) Gamma[\[Alpha], (-2 I c k + I c m + 2 a s - a v) z])/ ((-2 I c k + I c m + 2 a s - a v) z)^\[Alpha] - (E^(2 I d m) Gamma[\[Alpha], (2 I c k - I c m - 2 a s + a v) z])/ ((2 I c k - I c m - 2 a s + a v) z)^\[Alpha] - (E^(4 I d k) Gamma[\[Alpha], (-2 I c k + I c m - 2 a s + a v) z])/ ((-2 I c k + I c m - 2 a s + a v) z)^\[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 cos m ( d + c z ) cosh v ( a z ) z - 2 - m - v ( 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 - v mod 2 \$CellContext`v 2 ) z α k = 0 m - 1 2 - 2 d k - d m ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 d m Γ ( α , - c ( m - 2 k ) z ) ( - c ( m - 2 k ) z ) - α + 4 d k ( ( c m - 2 c k ) z ) - α Γ ( α , ( c m - 2 c k ) z ) ) - 2 - m - v ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) z α s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( α , - a ( v - 2 s ) z ) ( - a ( v - 2 s ) z ) - α + ( ( a v - 2 a s ) z ) - α Γ ( α , ( a v - 2 a s ) z ) ) + 2 - m - v z α k = 0 m - 1 2 ( 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 - 2 d k - d m ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 4 d k 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 - 2 d m ( ( 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 ) - 4 d k ( ( - 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 d m ( ( 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 - m - v 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]]]]] ( 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 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) α /; m + v + Condition z z α -1 d c z m a z v -1 2 -1 m -1 v Binomial v v 2 -1 1 -1 \$CellContext`v 2 z α k 0 m -1 2 -1 -2 d k -1 d m Binomial m k 2 d m Gamma α -1 c m -1 2 k z -1 c m -1 2 k z -1 α 4 d k c m -1 2 c k z -1 α Gamma α c m -1 2 c k z -1 2 -1 m -1 v Binomial m m 2 -1 1 -1 \$CellContext`m 2 z α s 0 v -1 2 -1 Binomial v s Gamma α -1 a v -1 2 s z -1 a v -1 2 s z -1 α a v -1 2 a s z -1 α Gamma α a v -1 2 a s z 2 -1 m -1 v z α k 0 m -1 2 -1 Binomial m k s 0 v -1 2 -1 -2 d k -1 d m Binomial v s -1 4 d k Gamma α -2 c k c m -1 2 a s a v z -2 c k c m -1 2 a s a v z -1 α -1 2 d m 2 c k -1 c m -1 2 a s a v z -1 α Gamma α 2 c k -1 c m -1 2 a s a v z -1 4 d k -2 c k c m 2 a s -1 a v z -1 α Gamma α -2 c k c m 2 a s -1 a v z -1 2 d m 2 c k -1 c m 2 a s -1 a v z -1 α Gamma α 2 c k -1 c m 2 a s -1 a v z 2 -1 m -1 v z α Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 α -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18