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/01.20.21.0369.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]]]]]]], "-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", 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RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]]]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]]]]]]], "-", "g"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]], RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18