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 Cosh

 http://functions.wolfram.com/01.20.21.4528.01

 Input Form

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