html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.4467.01

 Input Form

 Integrate[z^n Sinh[b z] Cosh[c z^2]^v, z] == 2^(-2 - v) (2 b^(-1 - n) Binomial[v, v/2] ((-1)^n Gamma[1 + n, (-b) z] + Gamma[1 + n, b z]) (1 - Mod[v, 2]) - c^(-1 - n) Sum[((-2 s + v)^(-1 - n) Binomial[v, s] ((-E^(b^2/(c (4 s - 2 v)))) Sum[2^(-n + q) b^(n - q) ((b + 4 c s z - 2 c v z)^2/(2 c s - c v))^((1/2) (-1 - q)) (-b + 2 c (-2 s + v) z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b + 4 c s z - 2 c v z)^2/(c (8 s - 4 v))], {q, 0, n}] - (-1)^n Sum[2^(-n + q) (-b)^(n - q) (b + 4 c s z - 2 c v z)^(1 + q) (-((b + 4 c s z - 2 c v z)^2/(2 c s - c v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((b + 4 c s z - 2 c v z)^2/( 8 c s - 4 c v))], {q, 0, n}] + (-1)^n Sum[2^(-n + q) b^(n - q) (-b + 4 c s z - 2 c v z)^(1 + q) (-((b + 2 c (-2 s + v) z)^2/(c (2 s - v))))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((b - 4 c s z + 2 c v z)^2/( 8 c s - 4 c v))], {q, 0, n}] - E^(b^2/(4 c s - 2 c v)) Sum[2^(-n + q) (-b)^(n - q) (-b + 4 c s z - 2 c v z) (b + 2 c (-2 s + v) z)^q ((b + 2 c (-2 s + v) z)^2/(c (2 s - v)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (b - 4 c s z + 2 c v z)^2/(8 c s - 4 c v)], {q, 0, n}]))/ E^(b^2/(c (8 s - 4 v))), {s, 0, Floor[(1/2) (-1 + v)]}]) /; Element[n, Integers] && n >= 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sinh ( b z ) cosh v ( c z 2 ) z 2 - v - 2 ( 2 b - n - 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 ) n Γ ( n + 1 , - b z ) + Γ ( n + 1 , b z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) - c - n - 1 s = 0 v - 1 2 - b 2 c ( 8 s - 4 v ) ( v - 2 s ) - n - 1 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - b 2 c ( 4 s - 2 v ) q = 0 n 2 q - n b n - q ( ( b + 4 c s z - 2 c v z ) 2 2 c s - c v ) 1 2 ( - q - 1 ) ( 2 c ( v - 2 s ) z - b ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b + 4 c s z - 2 c v z ) 2 c ( 8 s - 4 v ) ) - b 2 4 c s - 2 c v q = 0 n 2 q - n ( - b ) n - q ( - b + 4 c s z - 2 c v z ) ( b + 2 c ( v - 2 s ) z ) q ( ( b + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b - 4 c s z + 2 c v z ) 2 8 c s - 4 c v ) + ( - 1 ) n q = 0 n 2 q - n b n - q ( - b + 4 c s z - 2 c v z ) q + 1 ( - ( b + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( b - 4 c s z + 2 c v z ) 2 8 c s - 4 c v ) - ( - 1 ) n q = 0 n 2 q - n ( - b ) n - q ( b + 4 c s z - 2 c v z ) q + 1 ( - ( b + 4 c s z - 2 c v z ) 2 2 c s - c v ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( b + 4 c s z - 2 c v z ) 2 8 c s - 4 c v ) ) ) /; n v + Condition z z n b z c z 2 v 2 -1 v -2 2 b -1 n -1 Binomial v v 2 -1 -1 n Gamma n 1 -1 b z Gamma n 1 b z 1 -1 \$CellContext`v 2 -1 c -1 n -1 s 0 v -1 2 -1 -1 b 2 c 8 s -1 4 v -1 v -1 2 s -1 n -1 Binomial v s -1 b 2 c 4 s -1 2 v -1 q 0 n 2 q -1 n b n -1 q b 4 c s z -1 2 c v z 2 2 c s -1 c v -1 1 2 -1 q -1 2 c v -1 2 s z -1 b q 1 Binomial n q Gamma q 1 2 -1 b 4 c s z -1 2 c v z 2 c 8 s -1 4 v -1 -1 b 2 4 c s -1 2 c v -1 q 0 n 2 q -1 n -1 b n -1 q -1 b 4 c s z -1 2 c v z b 2 c v -1 2 s z q b 2 c v -1 2 s z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 b -1 4 c s z 2 c v z 2 8 c s -1 4 c v -1 -1 n q 0 n 2 q -1 n b n -1 q -1 b 4 c s z -1 2 c v z q 1 -1 b 2 c v -1 2 s z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b -1 4 c s z 2 c v z 2 8 c s -1 4 c v -1 -1 -1 n q 0 n 2 q -1 n -1 b n -1 q b 4 c s z -1 2 c v z q 1 -1 b 4 c s z -1 2 c v z 2 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b 4 c s z -1 2 c v z 2 8 c s -1 4 c v -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18