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 Cosh

 http://functions.wolfram.com/01.20.21.4650.01

 Input Form

 Integrate[E^(p Sqrt[z]) Sinh[b Sqrt[z]]^m Cosh[c z]^v, z] == (1/p^2) ((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^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (-((2 E^(p Sqrt[z] - c (-2 s + v) z))/ (c (-2 s + v))) + (2 E^(p Sqrt[z] + c (-2 s + v) z))/ (c (-2 s + v)) - (E^(p^2/(4 c (-2 s + v))) p Sqrt[Pi] Erfi[(p - 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[(-c) (-2 s + v)])])/ ((-c) (-2 s + v))^(3/2) - (p Sqrt[Pi] Erfi[(p + 2 c (-2 s + v) Sqrt[z])/ (2 Sqrt[c (-2 s + v)])])/(E^(p^2/(4 c (-2 s + v))) (c (-2 s + v))^(3/2))), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m + (2^(2 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k E^(p Sqrt[z]) Binomial[m, k] ((-p^2 - b^2 (-2 k + m)^2 (1 + p Sqrt[z]) + p^3 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 k + m) - p)^2 ((-b) (-2 k + m) + p)^2), {k, 0, Floor[(1/2) (-1 + m)]}])/I^m + 2^(-1 - m - v) (-1)^m Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (-((2 E^((b (2 k - m) + p) Sqrt[z] - c (-2 s + v) z))/(c (-2 s + v))) + (2 E^(I m Pi + ((-b) (2 k - m) + p) Sqrt[z] + c (-2 s + v) z))/ (c (-2 s + v)) + (E^((2 I b k - I b m + I p)^2/(c (8 s - 4 v))) (2 b k - b m + p) Sqrt[Pi] Erfi[((-b) (2 k - m) - p + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (2 s - v)])])/ (c (2 s - v))^(3/2) + (E^(I m Pi - (I b (-2 k + m) + I p)^2/ (c (8 s - 4 v))) ((-b) (2 k - m) + p) Sqrt[Pi] Erfi[((-b) (2 k - m) + p + 2 c (-2 s + v) Sqrt[z])/ (2 Sqrt[c (-2 s + v)])])/(c (2 s - v) Sqrt[c (-2 s + v)]) + (-1)^m (-((2 E^((b (-2 k + m) + p) Sqrt[z] - c (-2 s + v) z))/ (c (-2 s + v))) + (2 E^((-I) m Pi + ((-b) (-2 k + m) + p) Sqrt[z] + c (-2 s + v) z))/(c (-2 s + v)) + (E^((I b (-2 k + m) + I p)^2/(c (8 s - 4 v))) ((-b) (2 k - m) + p) Sqrt[Pi] Erfi[((-b) (-2 k + m) - p + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (2 s - v)])])/(c (2 s - v))^(3/2) + (E^((-I) m Pi - ((-b) (-2 k + m) + p)^2/(4 c (-2 s + v))) (2 b k - b m + p) Sqrt[Pi] Erfi[((-b) (-2 k + m) + p + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (-2 s + v)])])/ (c (2 s - v) Sqrt[c (-2 s + v)]))), {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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RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SqrtBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 p z sinh m ( b z ) cosh v ( c z ) z 2 - m - v + 1 - m 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 + 2 - m - v - 1 - m ( 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 ) 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 4 c ( v - 2 s ) p π erfi ( p - 2 c ( v - 2 s ) z 2 - c ( v - 2 s ) ) ( - c ( v - 2 s ) ) 3 / 2 + 2 z p + c ( v - 2 s ) z c ( v - 2 s ) - 2 p z - c ( v - 2 s ) z c ( v - 2 s ) - - p 2 4 c ( v - 2 s ) p π erfi ( p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 3 / 2 ) + 2 - m - v + 2 - m ( 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 ) k = 0 m - 1 2 ( ( - 1 ) k p z ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( z p 3 - p 2 - b 2 ( m - 2 k ) 2 ( z p + 1 ) ) cos ( m π 2 - b ( m - 2 k ) z ) + b ( m - 2 k ) ( b 2 z ( m - 2 k ) 2 + 2 p - p 2 z ) sin ( m π 2 - b ( m - 2 k ) z ) ) ) / ( ( - b ( m - 2 k ) - p ) 2 ( p - b ( m - 2 k ) ) 2 ) + 2 - m - v - 1 ( - 1 ) 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 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m ( ( b ( m - 2 k ) + p ) 2 c ( 8 s - 4 v ) ( p - b ( 2 k - m ) ) π erfi ( - b ( m - 2 k ) - p + 2 c ( v - 2 s ) z 2 c ( 2 s - v ) ) ( c ( 2 s - v ) ) 3 / 2 + ( - ( p - b ( m - 2 k ) ) 2 4 c ( v - 2 s ) - m π ( 2 b k - b m + p ) π erfi ( - b ( m - 2 k ) + p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ) / ( c ( 2 s - v ) c ( v - 2 s ) ) + 2 - π m + c ( v - 2 s ) z + ( p - b ( m - 2 k ) ) z c ( v - 2 s ) - 2 ( b ( m - 2 k ) + p ) z - c ( v - 2 s ) z c ( v - 2 s ) ) + ( 2 b k - b m + p ) 2 c ( 8 s - 4 v ) ( 2 b k - b m + p ) π erfi ( - b ( 2 k - m ) - p + 2 c ( v - 2 s ) z 2 c ( 2 s - v ) ) ( c ( 2 s - v ) ) 3 / 2 + m π - ( b ( m - 2 k ) + p ) 2 c ( 8 s - 4 v ) ( p - b ( 2 k - m ) ) π erfi ( - b ( 2 k - m ) + p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) c ( 2 s - v ) c ( v - 2 s ) + 2 π m + c ( v - 2 s ) z + ( p - b ( 2 k - m ) ) z c ( v - 2 s ) - 2 ( b ( 2 k - m ) + p ) z - c ( v - 2 s ) z c ( v - 2 s ) ) /; 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 v 2 -1 m -1 v 1 -1 m 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 2 -1 m -1 v -1 -1 m Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s -1 p 2 4 c v -1 2 s -1 p 1 2 Erfi p -1 2 c v -1 2 s z 1 2 2 -1 c v -1 2 s 1 2 -1 -1 c v -1 2 s 3 2 -1 2 z 1 2 p c v -1 2 s z c v -1 2 s -1 -1 2 p z 1 2 -1 c v -1 2 s z c v -1 2 s -1 -1 -1 p 2 4 c v -1 2 s -1 p 1 2 Erfi p 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c v -1 2 s 3 2 -1 2 -1 m -1 v 2 -1 m Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k p z 1 2 Binomial m k z 1 2 p 3 -1 p 2 -1 b 2 m -1 2 k 2 z 1 2 p 1 m 2 -1 -1 b m -1 2 k z 1 2 b m -1 2 k b 2 z 1 2 m -1 2 k 2 2 p -1 p 2 z 1 2 m 2 -1 -1 b m -1 2 k z 1 2 -1 b m -1 2 k -1 p 2 p -1 b m -1 2 k 2 -1 2 -1 m -1 v -1 -1 m k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 m b m -1 2 k p 2 c 8 s -1 4 v -1 p -1 b 2 k -1 m 1 2 Erfi -1 b m -1 2 k -1 p 2 c v -1 2 s z 1 2 2 c 2 s -1 v 1 2 -1 c 2 s -1 v 3 2 -1 -1 p -1 b m -1 2 k 2 4 c v -1 2 s -1 -1 m 2 b k -1 b m p 1 2 Erfi -1 b m -1 2 k p 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c 2 s -1 v c v -1 2 s 1 2 -1 2 -1 m c v -1 2 s z p -1 b m -1 2 k z 1 2 c v -1 2 s -1 -1 2 b m -1 2 k p z 1 2 -1 c v -1 2 s z c v -1 2 s -1 2 b k -1 b m p 2 c 8 s -1 4 v -1 2 b k -1 b m p 1 2 Erfi -1 b 2 k -1 m -1 p 2 c v -1 2 s z 1 2 2 c 2 s -1 v 1 2 -1 c 2 s -1 v 3 2 -1 m -1 b m -1 2 k p 2 c 8 s -1 4 v -1 p -1 b 2 k -1 m 1 2 Erfi -1 b 2 k -1 m p 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c 2 s -1 v c v -1 2 s 1 2 -1 2 m c v -1 2 s z p -1 b 2 k -1 m z 1 2 c v -1 2 s -1 -1 2 b 2 k -1 m p z 1 2 -1 c v -1 2 s z c v -1 2 s -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