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/01.20.21.0362.01

 Input Form

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

 Standard Form

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 MathML Form

 z n z b + d z cosh ( c z ) z 2 - 2 n - 2 ( - b 2 4 ( d - c ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k b - h - k + 2 n ( b + 2 ( d - c ) z ) h + k ( - ( b + 2 ( d - c ) z ) 2 d - c ) 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 ( d - c ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 ( d - c ) z ) 2 4 ( d - c ) ) + 2 - ( b + 2 ( d - c ) z ) 2 d - c ( d - c ) Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 ( d - c ) z ) 2 4 ( d - c ) ) ) ) ( d - c ) - 2 n - 2 + ( c + d ) - 2 n - 2 - b 2 4 ( c + d ) k = 0 n h = 0 k ( - 1 ) k - h 4 k b - h - k + 2 n ( b + 2 ( c + d ) z ) h + k ( - ( b + 2 ( c + d ) z ) 2 c + d ) 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 + d ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 ( c + d ) z ) 2 4 ( c + d ) ) + 2 - ( b + 2 ( c + d ) z ) 2 c + d ( c + d ) Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 ( c + d ) z ) 2 4 ( c + d ) ) ) ) /; n Condition z z n z 1 2 b d z c z 2 -2 n -2 -1 b 2 4 d -1 c -1 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 d -1 c z 1 2 h k -1 b 2 d -1 c z 1 2 2 d -1 c -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 d -1 c z 1 2 Gamma 1 2 h k 1 -1 b 2 d -1 c z 1 2 2 4 d -1 c -1 2 -1 b 2 d -1 c z 1 2 2 d -1 c -1 1 2 d -1 c Gamma 1 2 h k 2 -1 b 2 d -1 c z 1 2 2 4 d -1 c -1 d -1 c -2 n -2 c d -2 n -2 -1 b 2 4 c d -1 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 c d z 1 2 h k -1 b 2 c d z 1 2 2 c d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 c d z 1 2 Gamma 1 2 h k 1 -1 b 2 c d z 1 2 2 4 c d -1 2 -1 b 2 c d z 1 2 2 c d -1 1 2 c d Gamma 1 2 h k 2 -1 b 2 c d z 1 2 2 4 c d -1 n [/itex]

 Rule Form

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

 2002-12-18