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 Cosh

 http://functions.wolfram.com/01.20.21.4149.01

 Input Form

 Integrate[Sinh[b z^2]^m Cosh[c z]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) - (1/b) (2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/(-2 k + m)) ((-1)^k Binomial[m, k] ((-1)^m Sqrt[b (2 k - m)] Erfi[Sqrt[b (2 k - m)] z] + Sqrt[b (-2 k + m)] Erfi[(b (2 k - m) z)/Sqrt[b (-2 k + m)]])), {k, 0, Floor[(1/2) (-1 + m)]}]) - (1/c) (I^m 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, s] Sinh[(2 s - v) c z])/(-2 s + v), {s, 0, Floor[(1/2) (-1 + v)]}]) + 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (-(((-1)^m Sqrt[2 b k - b m] Erfi[(c (2 s - v) + 4 b k z - 2 b m z)/ (2 Sqrt[2 b k - b m])])/(E^((-2 c s + c v)^2/(8 b k - 4 b m)) (b (-2 k + m)))) + (E^((-2 c s + c v)^2/(8 b k - 4 b m)) Erfi[((-c) (2 s - v) - 4 b k z + 2 b m z)/(2 Sqrt[b (-2 k + m)])])/ Sqrt[b (-2 k + m)] + (1/(-2 b k + b m)) (((-(-1)^m) Sqrt[2 b k - b m] Erfi[((-c) (2 s - v) + 4 b k z - 2 b m z)/(2 Sqrt[2 b k - b m])] + E^((-2 c s + c v)^2/(4 b k - 2 b m)) Sqrt[-2 b k + b m] Erfi[(c (2 s - v) - 4 b k z + 2 b m z)/(2 Sqrt[-2 b k + b m])])/ E^((c^2 (-2 s + v)^2)/(8 b k - 4 b m)))), {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

 sinh m ( b z 2 ) cosh v ( c z ) z m 2 - m - v 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 ) - 2 - m - v - 1 π ( 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 ) b 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]]]]] ( b ( m - 2 k ) erfi ( b ( 2 k - m ) z b ( m - 2 k ) ) + ( - 1 ) m b ( 2 k - m ) erfi ( b ( 2 k - m ) z ) ) ) - m 2 - m - v + 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) c 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]]]]] sinh ( ( 2 s - v ) c z ) v - 2 s + 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]]]]] ( ( c v - 2 c s ) 2 8 b k - 4 b m erfi ( - c ( 2 s - v ) - 4 b k z + 2 b m z 2 b ( m - 2 k ) ) b ( m - 2 k ) + 1 b m - 2 b k ( - c 2 ( v - 2 s ) 2 8 b k - 4 b m ( ( c v - 2 c s ) 2 4 b k - 2 b m b m - 2 b k erfi ( c ( 2 s - v ) - 4 b k z + 2 b m z 2 b m - 2 b k ) - ( - 1 ) m 2 b k - b m erfi ( - c ( 2 s - v ) + 4 b k z - 2 b m z 2 2 b k - b m ) ) ) - ( - 1 ) m - ( c v - 2 c s ) 2 8 b k - 4 b m 2 b k - b m erfi ( c ( 2 s - v ) + 4 b k z - 2 b m z 2 2 b k - b m ) b ( m - 2 k ) ) /; m + v + Condition z b z 2 m c z v m 2 -1 m -1 v z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 -1 2 -1 m -1 v -1 1 2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 b -1 k 0 m -1 2 -1 1 m -1 2 k -1 -1 k Binomial m k b m -1 2 k 1 2 Erfi b 2 k -1 m z b m -1 2 k 1 2 -1 -1 m b 2 k -1 m 1 2 Erfi b 2 k -1 m 1 2 z -1 m 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c -1 s 0 v -1 2 -1 Binomial v s 2 s -1 v c z v -1 2 s -1 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 c v -1 2 c s 2 8 b k -1 4 b m -1 Erfi -1 c 2 s -1 v -1 4 b k z 2 b m z 2 b m -1 2 k 1 2 -1 b m -1 2 k 1 2 -1 1 b m -1 2 b k -1 -1 c 2 v -1 2 s 2 8 b k -1 4 b m -1 c v -1 2 c s 2 4 b k -1 2 b m -1 b m -1 2 b k 1 2 Erfi c 2 s -1 v -1 4 b k z 2 b m z 2 b m -1 2 b k 1 2 -1 -1 -1 m 2 b k -1 b m 1 2 Erfi -1 c 2 s -1 v 4 b k z -1 2 b m z 2 2 b k -1 b m 1 2 -1 -1 -1 m -1 c v -1 2 c s 2 8 b k -1 4 b m -1 2 b k -1 b m 1 2 Erfi c 2 s -1 v 4 b k z -1 2 b m z 2 2 b k -1 b m 1 2 -1 b m -1 2 k -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18