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/01.20.21.0518.01

 Input Form

 Integrate[z^n E^(b Sqrt[z]) Cosh[c Sqrt[z] + f z + g], z] == 2^(-2 - 2 n) f^(-2 - 2 n) (E^((b - c)^2/(4 f) - g) Sum[(-1)^(-h + k) 4^k (b - c)^(-h - k + 2 n) (b - c - 2 f Sqrt[z])^ (h + k) ((b - c - 2 f Sqrt[z])^2/f)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b - c) (b - c - 2 f Sqrt[z]) Gamma[(1/2) (1 + h + k), (b - c - 2 f Sqrt[z])^2/(4 f)] - 2 f Sqrt[(b - c - 2 f Sqrt[z])^2/f] Gamma[(1/2) (2 + h + k), (b - c - 2 f Sqrt[z])^2/(4 f)]), {k, 0, n}, {h, 0, k}] + E^(-((b + c)^2/(4 f)) + g) Sum[(-1)^(-h + k) 4^k (b + c)^(-h - k + 2 n) (b + c + 2 f Sqrt[z])^(h + k) (-((b + c + 2 f Sqrt[z])^2/f))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b + c) (b + c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + k), -((b + c + 2 f Sqrt[z])^2/(4 f))] + 2 f Sqrt[-((b + c + 2 f Sqrt[z])^2/f)] Gamma[(1/2) (2 + h + k), -((b + c + 2 f Sqrt[z])^2/(4 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["b", " ", SqrtBox["z"]]]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]], "-", "g"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "+", "g"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n z b cosh ( z c + g + f z ) z 2 - 2 n - 2 f - 2 n - 2 ( ( b - c ) 2 4 f - g k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b - c ) - h - k + 2 n ( b - c - 2 f z ) h + k ( ( b - c - 2 f z ) 2 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 - c ) ( b - c - 2 f z ) Γ ( 1 2 ( h + k + 1 ) , ( b - c - 2 f z ) 2 4 f ) - 2 f ( b - c - 2 f z ) 2 f Γ ( 1 2 ( h + k + 2 ) , ( b - c - 2 f z ) 2 4 f ) ) + - ( b + c ) 2 4 f + g k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + c ) - h - k + 2 n ( b + c + 2 f z ) h + k ( - ( b + c + 2 f z ) 2 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 + c ) ( b + c + 2 f z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + c + 2 f z ) 2 4 f ) + 2 - ( b + c + 2 f z ) 2 f f Γ ( 1 2 ( h + k + 2 ) , - ( b + c + 2 f z ) 2 4 f ) ) ) /; n Condition z z n z 1 2 b z 1 2 c g f z 2 -2 n -2 f -2 n -2 b -1 c 2 4 f -1 -1 g h 0 k k 0 n -1 k -1 h 4 k b -1 c -1 h -1 k 2 n b -1 c -1 2 f z 1 2 h k b -1 c -1 2 f z 1 2 2 f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b -1 c b -1 c -1 2 f z 1 2 Gamma 1 2 h k 1 b -1 c -1 2 f z 1 2 2 4 f -1 -1 2 f b -1 c -1 2 f z 1 2 2 f -1 1 2 Gamma 1 2 h k 2 b -1 c -1 2 f z 1 2 2 4 f -1 -1 b c 2 4 f -1 g h 0 k k 0 n -1 k -1 h 4 k b c -1 h -1 k 2 n b c 2 f z 1 2 h k -1 b c 2 f z 1 2 2 f -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b c b c 2 f z 1 2 Gamma 1 2 h k 1 -1 b c 2 f z 1 2 2 4 f -1 2 -1 b c 2 f z 1 2 2 f -1 1 2 f Gamma 1 2 h k 2 -1 b c 2 f z 1 2 2 4 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["b_", " ", SqrtBox["z_"]]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", 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"]]]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]], "-", "g"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "+", "g"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18