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/01.20.21.0501.01

 Input Form

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

 MathML Form

 z n d z cosh ( c z 2 + f z ) z - 1 4 ( 1 - c ( d - f ) 2 4 c q = 0 n 2 q - n ( - c ) - n - 1 2 ( f - d ) n - q ( d - f - 2 c z ) q + 1 ( ( d - f - 2 c z ) 2 c ) 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 , ( d - f - 2 c z ) 2 4 c ) + 1 c ( - ( d + f ) 2 4 c q = 0 n 2 q - n c - n - 1 2 ( - d - f ) n - q ( d + f + 2 c z ) q + 1 ( - ( d + f + 2 c z ) 2 c ) 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 , - ( d + f + 2 c z ) 2 4 c ) ) ) /; n Condition z z n d z c z 2 f z -1 1 4 1 -1 c 1 2 -1 d -1 f 2 4 c -1 q 0 n 2 q -1 n -1 c -1 n -1 1 2 f -1 d n -1 q d -1 f -1 2 c z q 1 d -1 f -1 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d -1 f -1 2 c z 2 4 c -1 1 c 1 2 -1 -1 d f 2 4 c -1 q 0 n 2 q -1 n c -1 n -1 1 2 -1 d -1 f n -1 q d f 2 c z q 1 -1 d f 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d f 2 c z 2 4 c -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["d_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f"]], ")"]], "2"], RowBox[List["4", " ", "c"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "c"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", "f"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f", "-", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f", "-", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], "2"], "c"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "-", "f", "-", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]], "]"]]]]]]]], SqrtBox[RowBox[List["-", "c"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f"]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", "f"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], "2"], "c"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", "f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", "c"]]]]]]], "]"]]]]]]]], SqrtBox["c"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18