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/01.20.21.0936.01

 Input Form

 Integrate[z^n Sin[b z^2 + e] Cosh[c z^2 + f z], z] == (-(1/8)) I ((((-I) b - c)^(-1 - n) Sum[2^(j - n) f^(-j + n) (-f + 2 ((-I) b - c) z)^(1 + j) (-((-f + 2 ((-I) b - c) z)^2/((-I) b - c)))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((-f + 2 ((-I) b - c) z)^2/ (4 ((-I) b - c)))], {j, 0, n}])/ E^((1/4) I (4 e + f^2/(b - I c))) - (I b - c)^(-1 - n) E^((1/4) I (4 e - f^2/(-b - I c))) Sum[2^(j - n) f^(-j + n) (-f + 2 (I b - c) z)^(1 + j) (-((-f + 2 (I b - c) z)^2/(I b - c)))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((-f + 2 (I b - c) z)^2/(4 (I b - c)))], {j, 0, n}] + (((-I) b + c)^(-1 - n) Sum[2^(j - n) (-f)^(-j + n) (f + 2 ((-I) b + c) z)^(1 + j) (-((f + 2 ((-I) b + c) z)^2/((-I) b + c)))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((f + 2 ((-I) b + c) z)^2/ (4 ((-I) b + c)))], {j, 0, n}])/ E^((1/4) I (4 e - f^2/(-b - I c))) - (I b + c)^(-1 - n) E^((1/4) I (4 e + f^2/(b - I c))) Sum[2^(j - n) (-f)^(-j + n) (f + 2 (I b + c) z)^(1 + j) (-((f + 2 (I b + c) z)^2/(I b + c)))^ ((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((f + 2 (I b + c) z)^2/(4 (I b + c)))], {j, 0, n}]) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], RowBox[List["Sin", "[", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", "e"]], "]"]], 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", "8"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]]]]]]]]], "]"]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]]]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]]]]]]]]], "]"]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]]]]]]]]], "]"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n sin ( b z 2 + e ) cosh ( c z 2 + f z ) z - 1 8 ( - 1 4 ( f 2 b - c + 4 e ) ( j = 0 n 2 j - n f n - j ( 2 ( - c - b ) z - f ) j + 1 ( - ( 2 ( - c - b ) z - f ) 2 - c - b ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( 2 ( - c - b ) z - f ) 2 4 ( - c - b ) ) ) ( - c - b ) - n - 1 - ( b - c ) - n - 1 1 4 ( - f 2 - b - c + 4 e ) j = 0 n 2 j - n f n - j ( 2 ( b - c ) z - f ) j + 1 ( - ( 2 ( b - c ) z - f ) 2 b - c ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( 2 ( b - c ) z - f ) 2 4 ( b - c ) ) + ( c - b ) - n - 1 - 1 4 ( - f 2 - b - c + 4 e ) j = 0 n 2 j - n ( - f ) n - j ( f + 2 ( c - b ) z ) j + 1 ( - ( f + 2 ( c - b ) z ) 2 c - b ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( f + 2 ( c - b ) z ) 2 4 ( c - b ) ) - ( c + b ) - n - 1 1 4 ( f 2 b - c + 4 e ) j = 0 n 2 j - n ( - f ) n - j ( f + 2 ( c + b ) z ) j + 1 ( - ( f + 2 ( c + b ) z ) 2 c + b ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( f + 2 ( c + b ) z ) 2 4 ( c + b ) ) ) /; n Condition z z n b z 2 e c z 2 f z -1 1 8 -1 1 4 f 2 b -1 c -1 4 e j 0 n 2 j -1 n f n -1 j 2 -1 c -1 b z -1 f j 1 -1 2 -1 c -1 b z -1 f 2 -1 c -1 b -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 2 -1 c -1 b z -1 f 2 4 -1 c -1 b -1 -1 c -1 b -1 n -1 -1 b -1 c -1 n -1 1 4 -1 f 2 -1 b -1 c -1 4 e j 0 n 2 j -1 n f n -1 j 2 b -1 c z -1 f j 1 -1 2 b -1 c z -1 f 2 b -1 c -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 2 b -1 c z -1 f 2 4 b -1 c -1 c -1 b -1 n -1 -1 1 4 -1 f 2 -1 b -1 c -1 4 e j 0 n 2 j -1 n -1 f n -1 j f 2 c -1 b z j 1 -1 f 2 c -1 b z 2 c -1 b -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 f 2 c -1 b z 2 4 c -1 b -1 -1 c b -1 n -1 1 4 f 2 b -1 c -1 4 e j 0 n 2 j -1 n -1 f n -1 j f 2 c b z j 1 -1 f 2 c b z 2 c b -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 f 2 c b z 2 4 c b -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", "e_"]], "]"]], " ", 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", "8"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]]]]]]]]], "]"]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]]]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "-", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]]]]]]]]], "]"]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "e"]], "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "f"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]]]]]]]]], "]"]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18