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 Cosh

 http://functions.wolfram.com/01.20.21.4661.01

 Input Form

 Integrate[E^(p z) Sinh[b z^2]^m Cosh[c z^2]^v, z] == (1/p) (I^m 2^(-m - v) E^(p z) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) - ((I^m 2^(-1 - m - v))/b) Binomial[v, v/2] (1 - Mod[v, 2]) Sqrt[Pi] Sum[(1/(-2 k + m)) ((-1)^k Binomial[m, k] (E^((p^2 + 2 I b m (-2 k + m) Pi)/(4 b (-2 k + m))) Sqrt[(-b) (-2 k + m)] Erfi[(p - 2 b (-2 k + m) z)/ (2 Sqrt[(-b) (-2 k + m)])] - (Sqrt[b (-2 k + m)] Erfi[(p + 2 b (-2 k + m) z)/(2 Sqrt[b (-2 k + m)])])/ E^((p^2 + 2 I b m (-2 k + m) Pi)/(4 b (-2 k + m))))), {k, 0, Floor[(1/2) (-1 + m)]}] - ((I^m 2^(-1 - m - v) Sqrt[Pi])/c) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(-2 s + v)) Binomial[v, s] (E^(p^2/(4 c (-2 s + v))) Sqrt[(-c) (-2 s + v)] Erfi[(p - 2 c (-2 s + v) z)/(2 Sqrt[(-c) (-2 s + v)])] - (Sqrt[c (-2 s + v)] Erfi[(p + 2 c (-2 s + v) z)/ (2 Sqrt[c (-2 s + v)])])/E^(p^2/(4 c (-2 s + v)))), {s, 0, Floor[(1/2) (-1 + v)]}] + I^m 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (((Sqrt[(-b) (2 k - m) - c (-2 s + v)] (b (2 k - m) + c (-2 s + v)) Erfi[(p - 2 (b (2 k - m) + c (-2 s + v)) z)/(2 Sqrt[(-b) (2 k - m) - c (-2 s + v)])])/ E^((p^2 + 2 I m Pi ((-b) (2 k - m) - c (-2 s + v)))/ (4 ((-b) (2 k - m) - c (-2 s + v)))) + (((-b) (2 k - m) - c (-2 s + v)) Sqrt[b (2 k - m) + c (-2 s + v)] Erfi[(p + 2 (b (2 k - m) + c (-2 s + v)) z)/(2 Sqrt[b (2 k - m) + c (-2 s + v)])])/ E^((p^2 - 2 I m Pi (b (2 k - m) + c (-2 s + v)))/ (4 (b (2 k - m) + c (-2 s + v)))))/ (((-b) (2 k - m) - c (-2 s + v)) (b (2 k - m) + c (-2 s + v))) + ((Sqrt[(-b) (-2 k + m) - c (-2 s + v)] (b (-2 k + m) + c (-2 s + v)) Erfi[(p - 2 (b (-2 k + m) + c (-2 s + v)) z)/(2 Sqrt[(-b) (-2 k + m) - c (-2 s + v)])])/ E^((p^2 - 2 I m Pi ((-b) (-2 k + m) - c (-2 s + v)))/ (4 ((-b) (-2 k + m) - c (-2 s + v)))) + (((-b) (-2 k + m) - c (-2 s + v)) Sqrt[b (-2 k + m) + c (-2 s + v)] Erfi[(p + 2 (b (-2 k + m) + c (-2 s + v)) z)/(2 Sqrt[b (-2 k + m) + c (-2 s + v)])])/ E^((p^2 + 2 I m Pi (b (-2 k + m) + c (-2 s + v)))/ (4 (b (-2 k + m) + c (-2 s + v)))))/ (((-b) (-2 k + m) - c (-2 s + v)) (b (-2 k + m) + 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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" ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", 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 2 ) cosh v ( c z 2 ) z m 2 - m - v p 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) p - 1 b ( ( m 2 - m - v - 1 ) ( 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 ) π k = 0 m - 1 2 1 m - 2 k ( ( - 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 + 2 b m ( m - 2 k ) π 4 b ( m - 2 k ) - b ( m - 2 k ) erfi ( p - 2 b ( m - 2 k ) z 2 - b ( m - 2 k ) ) - - p 2 + 2 b m ( m - 2 k ) π 4 b ( m - 2 k ) b ( m - 2 k ) erfi ( p + 2 b ( m - 2 k ) z 2 b ( m - 2 k ) ) ) ) ) - 1 c ( ( m 2 - m - v - 1 π ) ( 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 1 v - 2 s ( ( 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 ) - c ( v - 2 s ) erfi ( p - 2 c ( v - 2 s ) z 2 - c ( v - 2 s ) ) - - p 2 4 c ( v - 2 s ) c ( v - 2 s ) erfi ( p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ) ) ) + m 2 - m - v - 1 π 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 - 2 m π ( b ( 2 k - m ) + c ( v - 2 s ) ) 4 ( b ( 2 k - m ) + c ( v - 2 s ) ) b ( 2 k - m ) + c ( v - 2 s ) ( - b ( 2 k - m ) - c ( v - 2 s ) ) erfi ( p + 2 ( b ( 2 k - m ) + c ( v - 2 s ) ) z 2 b ( 2 k - m ) + c ( v - 2 s ) ) + - p 2 + 2 m π ( - b ( 2 k - m ) - c ( v - 2 s ) ) 4 ( - b ( 2 k - m ) - c ( v - 2 s ) ) ( b ( 2 k - m ) + c ( v - 2 s ) ) - b ( 2 k - m ) - c ( v - 2 s ) erfi ( p - 2 ( b ( 2 k - m ) + c ( v - 2 s ) ) z 2 - b ( 2 k - m ) - c ( v - 2 s ) ) ) / ( ( - b ( 2 k - m ) - c ( v - 2 s ) ) ( b ( 2 k - m ) + c ( v - 2 s ) ) ) + ( - p 2 + 2 m π ( b ( m - 2 k ) + c ( v - 2 s ) ) 4 ( b ( m - 2 k ) + c ( v - 2 s ) ) b ( m - 2 k ) + c ( v - 2 s ) ( - b ( m - 2 k ) - c ( v - 2 s ) ) erfi ( p + 2 ( b ( m - 2 k ) + c ( v - 2 s ) ) z 2 b ( m - 2 k ) + c ( v - 2 s ) ) + - p 2 - 2 m π ( - b ( m - 2 k ) - c ( v - 2 s ) ) 4 ( - b ( m - 2 k ) - c ( v - 2 s ) ) ( b ( m - 2 k ) + c ( v - 2 s ) ) - b ( m - 2 k ) - c ( v - 2 s ) erfi ( p - 2 ( b ( m - 2 k ) + c ( v - 2 s ) ) z 2 - b ( m - 2 k ) - c ( v - 2 s ) ) ) / ( ( - b ( m - 2 k ) - c ( v - 2 s ) ) ( b ( m - 2 k ) + c ( v - 2 s ) ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z b z 2 m c z 2 v m 2 -1 m -1 v p z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p -1 -1 1 b -1 m 2 -1 m -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 1 2 k 0 m -1 2 -1 1 m -1 2 k -1 -1 k Binomial m k p 2 2 b m m -1 2 k 4 b m -1 2 k -1 -1 b m -1 2 k 1 2 Erfi p -1 2 b m -1 2 k z 2 -1 b m -1 2 k 1 2 -1 -1 -1 p 2 2 b m m -1 2 k 4 b m -1 2 k -1 b m -1 2 k 1 2 Erfi p 2 b m -1 2 k z 2 b m -1 2 k 1 2 -1 -1 1 c -1 m 2 -1 m -1 v -1 1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 1 v -1 2 s -1 Binomial v s p 2 4 c v -1 2 s -1 -1 c v -1 2 s 1 2 Erfi p -1 2 c v -1 2 s z 2 -1 c v -1 2 s 1 2 -1 -1 -1 p 2 4 c v -1 2 s -1 c v -1 2 s 1 2 Erfi p 2 c v -1 2 s z 2 c v -1 2 s 1 2 -1 m 2 -1 m -1 v -1 1 2 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 p 2 -1 2 m b 2 k -1 m c v -1 2 s 4 b 2 k -1 m c v -1 2 s -1 b 2 k -1 m c v -1 2 s 1 2 -1 b 2 k -1 m -1 c v -1 2 s Erfi p 2 b 2 k -1 m c v -1 2 s z 2 b 2 k -1 m c v -1 2 s 1 2 -1 -1 p 2 2 m -1 b 2 k -1 m -1 c v -1 2 s 4 -1 b 2 k -1 m -1 c v -1 2 s -1 b 2 k -1 m c v -1 2 s -1 b 2 k -1 m -1 c v -1 2 s 1 2 Erfi p -1 2 b 2 k -1 m c v -1 2 s z 2 -1 b 2 k -1 m -1 c v -1 2 s 1 2 -1 -1 b 2 k -1 m -1 c v -1 2 s b 2 k -1 m c v -1 2 s -1 -1 p 2 2 m b m -1 2 k c v -1 2 s 4 b m -1 2 k c v -1 2 s -1 b m -1 2 k c v -1 2 s 1 2 -1 b m -1 2 k -1 c v -1 2 s Erfi p 2 b m -1 2 k c v -1 2 s z 2 b m -1 2 k c v -1 2 s 1 2 -1 -1 p 2 -1 2 m -1 b m -1 2 k -1 c v -1 2 s 4 -1 b m -1 2 k -1 c v -1 2 s -1 b m -1 2 k c v -1 2 s -1 b m -1 2 k -1 c v -1 2 s 1 2 Erfi p -1 2 b m -1 2 k c v -1 2 s z 2 -1 b m -1 2 k -1 c v -1 2 s 1 2 -1 -1 b m -1 2 k -1 c v -1 2 s b m -1 2 k 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