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 Cosh

 http://functions.wolfram.com/01.20.21.4663.01

 Input Form

 Integrate[E^(p Sqrt[z]) Sinh[b Sqrt[z]]^m Cosh[c Sqrt[z]]^v, z] == (1/p^2) (I^m 2^(1 - m - v) E^(p Sqrt[z]) (-1 + p Sqrt[z]) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + I^m 2^(2 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) E^(p Sqrt[z]) Sum[((-1)^k Binomial[m, k] ((p^2 (-1 + p Sqrt[z]) - b^2 (-2 k + m)^2 (1 + p Sqrt[z])) Cos[(m Pi)/2 + I b (-2 k + m) Sqrt[z]] + I b (-2 k + m) (-2 p - b^2 (-2 k + m)^2 Sqrt[z] + p^2 Sqrt[z]) Sin[(m Pi)/2 + I b (-2 k + m) Sqrt[z]]))/((-b^2) (-2 k + m)^2 + p^2)^ 2, {k, 0, Floor[(1/2) (-1 + m)]}] + I^m 2^(2 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) E^(p Sqrt[z]) Sum[(Binomial[v, s] ((p^2 (-1 + p Sqrt[z]) - c^2 (-2 s + v)^2 (1 + p Sqrt[z])) Cosh[c (-2 s + v) Sqrt[z]] - c (-2 s + v) (-2 p + p^2 Sqrt[z] - c^2 (-2 s + v)^2 Sqrt[z]) Sinh[c (-2 s + v) Sqrt[z]]))/(p^2 - c^2 (-2 s + v)^2)^2, {s, 0, Floor[(1/2) (-1 + v)]}] + I^m 2^(2 - m - v) E^(p Sqrt[z]) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (Sin[(m Pi)/2] (((-I) (2 b k - b m + 2 c s - c v) (-2 p + p^2 Sqrt[z] - (2 b k - b m + 2 c s - c v)^2 Sqrt[z]) Cosh[(2 b k - b m + 2 c s - c v) Sqrt[z]] + I (p^2 (-1 + p Sqrt[z]) - (2 b k - b m + 2 c s - c v)^2 (1 + p Sqrt[z])) Sinh[(2 b k - b m + 2 c s - c v) Sqrt[z]])/ (p^2 - (2 b k - b m + 2 c s - c v)^2)^2 + ((-I) (2 b k - b m - 2 c s + c v) (-2 p + p^2 Sqrt[z] - (2 b k - b m - 2 c s + c v)^2 Sqrt[z]) Cosh[(2 b k - b m - 2 c s + c v) Sqrt[z]] + I (p^2 (-1 + p Sqrt[z]) - (2 b k - b m - 2 c s + c v)^2 (1 + p Sqrt[z])) Sinh[(2 b k - b m - 2 c s + c v) Sqrt[z]])/ (p^2 - (2 b k - b m - 2 c s + c v)^2)^2) + Cos[(m Pi)/2] (((p^2 (-1 + p Sqrt[z]) - (2 b k - b m + 2 c s - c v)^ 2 (1 + p Sqrt[z])) Cosh[(2 b k - b m + 2 c s - c v) Sqrt[z]] - (2 b k - b m + 2 c s - c v) (-2 p + p^2 Sqrt[z] - (2 b k - b m + 2 c s - c v)^2 Sqrt[z]) Sinh[(2 b k - b m + 2 c s - c v) Sqrt[z]])/(p^2 - (2 b k - b m + 2 c s - c v)^2)^ 2 + ((p^2 (-1 + p Sqrt[z]) - (2 b k - b m - 2 c s + c v)^2 (1 + p Sqrt[z])) Cosh[(2 b k - b m - 2 c s + c v) Sqrt[z]] - (2 b k - b m - 2 c s + c v) (-2 p + p^2 Sqrt[z] - (2 b k - b m - 2 c s + c v)^2 Sqrt[z]) Sinh[(2 b k - b m - 2 c s + c v) Sqrt[z]])/(p^2 - (2 b k - b m - 2 c s + c v)^2)^ 2)), {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

 p z sinh m ( b z ) cosh v ( c z ) z m 2 - m - v + 1 p z ( p z - 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) p 2 + m 2 - m - v + 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 - v mod 2 \$CellContext`v 2 ) p z k = 0 m - 1 2 1 ( p 2 - b 2 ( m - 2 k ) 2 ) 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]]]]] ( ( p 2 ( p z - 1 ) - b 2 ( m - 2 k ) 2 ( z p + 1 ) ) cos ( π m 2 + b ( m - 2 k ) z ) + b ( m - 2 k ) ( - b 2 z ( m - 2 k ) 2 - 2 p + p 2 z ) sin ( π m 2 + b ( m - 2 k ) z ) ) ) + m 2 - m - v + 2 ( 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 ) p z s = 0 v - 1 2 1 ( p 2 - c 2 ( v - 2 s ) 2 ) 2 ( ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p 2 ( p z - 1 ) - c 2 ( v - 2 s ) 2 ( z p + 1 ) ) cosh ( c ( v - 2 s ) z ) - c ( v - 2 s ) ( z p 2 - 2 p - c 2 ( v - 2 s ) 2 z ) sinh ( c ( v - 2 s ) z ) ) ) + m 2 - m - v + 2 p 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 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( ( ( p 2 ( p z - 1 ) - ( 2 b k - b m - 2 c s + c v ) 2 ( z p + 1 ) ) cosh ( ( 2 b k - b m - 2 c s + c v ) z ) - ( 2 b k - b m - 2 c s + c v ) ( z p 2 - 2 p - ( 2 b k - b m - 2 c s + c v ) 2 z ) sinh ( ( 2 b k - b m - 2 c s + c v ) z ) ) / ( p 2 - ( 2 b k - b m - 2 c s + c v ) 2 ) 2 + ( ( p 2 ( p z - 1 ) - ( 2 b k - b m + 2 c s - c v ) 2 ( z p + 1 ) ) cosh ( ( 2 b k - b m + 2 c s - c v ) z ) - ( 2 b k - b m + 2 c s - c v ) ( z p 2 - 2 p - ( 2 b k - b m + 2 c s - c v ) 2 z ) sinh ( ( 2 b k - b m + 2 c s - c v ) z ) ) / ( p 2 - ( 2 b k - b m + 2 c s - c v ) 2 ) 2 ) cos ( m π 2 ) + ( ( ( p 2 ( p z - 1 ) - ( 2 b k - b m - 2 c s + c v ) 2 ( z p + 1 ) ) sinh ( ( 2 b k - b m - 2 c s + c v ) z ) - ( 2 b k - b m - 2 c s + c v ) ( z p 2 - 2 p - ( 2 b k - b m - 2 c s + c v ) 2 z ) cosh ( ( 2 b k - b m - 2 c s + c v ) z ) ) / ( p 2 - ( 2 b k - b m - 2 c s + c v ) 2 ) 2 + ( ( p 2 ( p z - 1 ) - ( 2 b k - b m + 2 c s - c v ) 2 ( z p + 1 ) ) sinh ( ( 2 b k - b m + 2 c s - c v ) z ) - ( 2 b k - b m + 2 c s - c v ) ( z p 2 - 2 p - ( 2 b k - b m + 2 c s - c v ) 2 z ) cosh ( ( 2 b k - b m + 2 c s - c v ) z ) ) / ( p 2 - ( 2 b k - b m + 2 c s - c v ) 2 ) 2 ) sin ( m π 2 ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z 1 2 b z 1 2 m c z 1 2 v m 2 -1 m -1 v 1 p z 1 2 p z 1 2 -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p 2 -1 m 2 -1 m -1 v 2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 p z 1 2 k 0 m -1 2 -1 1 p 2 -1 b 2 m -1 2 k 2 2 -1 -1 k Binomial m k p 2 p z 1 2 -1 -1 b 2 m -1 2 k 2 z 1 2 p 1 m 2 -1 b m -1 2 k z 1 2 b m -1 2 k -1 b 2 z 1 2 m -1 2 k 2 -1 2 p p 2 z 1 2 m 2 -1 b m -1 2 k z 1 2 m 2 -1 m -1 v 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 p z 1 2 s 0 v -1 2 -1 1 p 2 -1 c 2 v -1 2 s 2 2 -1 Binomial v s p 2 p z 1 2 -1 -1 c 2 v -1 2 s 2 z 1 2 p 1 c v -1 2 s z 1 2 -1 c v -1 2 s z 1 2 p 2 -1 2 p -1 c 2 v -1 2 s 2 z 1 2 c v -1 2 s z 1 2 m 2 -1 m -1 v 2 p z 1 2 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s p 2 p z 1 2 -1 -1 2 b k -1 b m -1 2 c s c v 2 z 1 2 p 1 2 b k -1 b m -1 2 c s c v z 1 2 -1 2 b k -1 b m -1 2 c s c v z 1 2 p 2 -1 2 p -1 2 b k -1 b m -1 2 c s c v 2 z 1 2 2 b k -1 b m -1 2 c s c v z 1 2 p 2 -1 2 b k -1 b m -1 2 c s c v 2 2 -1 p 2 p z 1 2 -1 -1 2 b k -1 b m 2 c s -1 c v 2 z 1 2 p 1 2 b k -1 b m 2 c s -1 c v z 1 2 -1 2 b k -1 b m 2 c s -1 c v z 1 2 p 2 -1 2 p -1 2 b k -1 b m 2 c s -1 c v 2 z 1 2 2 b k -1 b m 2 c s -1 c v z 1 2 p 2 -1 2 b k -1 b m 2 c s -1 c v 2 2 -1 m 2 -1 p 2 p z 1 2 -1 -1 2 b k -1 b m -1 2 c s c v 2 z 1 2 p 1 2 b k -1 b m -1 2 c s c v z 1 2 -1 2 b k -1 b m -1 2 c s c v z 1 2 p 2 -1 2 p -1 2 b k -1 b m -1 2 c s c v 2 z 1 2 2 b k -1 b m -1 2 c s c v z 1 2 p 2 -1 2 b k -1 b m -1 2 c s c v 2 2 -1 p 2 p z 1 2 -1 -1 2 b k -1 b m 2 c s -1 c v 2 z 1 2 p 1 2 b k -1 b m 2 c s -1 c v z 1 2 -1 2 b k -1 b m 2 c s -1 c v z 1 2 p 2 -1 2 p -1 2 b k -1 b m 2 c s -1 c v 2 z 1 2 2 b k -1 b m 2 c s -1 c v z 1 2 p 2 -1 2 b k -1 b m 2 c s -1 c v 2 2 -1 m 2 -1 m m 0 v v 0 [/itex]

 Rule Form

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

 2002-12-18