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/01.20.21.4454.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18