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/01.20.21.1085.01

 Input Form

 Integrate[z^n Cos[b Sqrt[z]] Cosh[c Sqrt[z] + f z], z] == (2^(-3 - 2 n) (E^(((-I) b - c)^2/(4 f)) Sum[(-1)^(-h + j) 4^j ((-I) b - c)^(-h - j + 2 n) ((-I) b - c - 2 f Sqrt[z])^(h + j) (((-I) b - c - 2 f Sqrt[z])^2/f)^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b - c) ((-I) b - c - 2 f Sqrt[z]) Gamma[(1/2) (1 + h + j), ((-I) b - c - 2 f Sqrt[z])^2/(4 f)] - 2 f Sqrt[((-I) b - c - 2 f Sqrt[z])^2/f] Gamma[(1/2) (2 + h + j), ((-I) b - c - 2 f Sqrt[z])^2/(4 f)]), {j, 0, n}, {h, 0, j}] + E^((I b - c)^2/(4 f)) Sum[(-1)^(-h + j) 4^j (I b - c)^(-h - j + 2 n) (I b - c - 2 f Sqrt[z])^(h + j) ((I b - c - 2 f Sqrt[z])^2/f)^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b - c) (I b - c - 2 f Sqrt[z]) Gamma[(1/2) (1 + h + j), (I b - c - 2 f Sqrt[z])^2/(4 f)] - 2 f Sqrt[(I b - c - 2 f Sqrt[z])^2/f] Gamma[(1/2) (2 + h + j), (I b - c - 2 f Sqrt[z])^2/(4 f)]), {j, 0, n}, {h, 0, j}] + Sum[(-1)^(-h + j) 4^j ((-I) b + c)^(-h - j + 2 n) ((-I) b + c + 2 f Sqrt[z])^(h + j) (-(((-I) b + c + 2 f Sqrt[z])^2/f))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (((-I) b + c) ((-I) b + c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b + c + 2 f Sqrt[z])^2/(4 f))] + 2 f Sqrt[-(((-I) b + c + 2 f Sqrt[z])^2/f)] Gamma[(1/2) (2 + h + j), -(((-I) b + c + 2 f Sqrt[z])^2/(4 f))]), {j, 0, n}, {h, 0, j}]/ E^(((-I) b + c)^2/(4 f)) + Sum[(-1)^(-h + j) 4^j (I b + c)^(-h - j + 2 n) (I b + c + 2 f Sqrt[z])^ (h + j) (-((I b + c + 2 f Sqrt[z])^2/f))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((I b + c) (I b + c + 2 f Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b + c + 2 f Sqrt[z])^2/(4 f))] + 2 f Sqrt[-((I b + c + 2 f Sqrt[z])^2/f)] Gamma[(1/2) (2 + h + j), -((I b + c + 2 f Sqrt[z])^2/(4 f))]), {j, 0, n}, {h, 0, j}]/ E^((I b + c)^2/(4 f))))/f^(2 (1 + n)) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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[RowBox[List["\[ImaginaryI]", " ", "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", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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 cos ( z b ) cosh ( z c + f z ) z 2 - 2 n - 3 f - 2 ( n + 1 ) ( ( - c - b ) 2 4 f j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c - b ) - h - j + 2 n ( - c - b - 2 f z ) h + j ( ( - c - b - 2 f z ) 2 f ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - c - b ) ( - c - b - 2 f z ) Γ ( 1 2 ( h + j + 1 ) , ( - c - b - 2 f z ) 2 4 f ) - 2 f ( - c - b - 2 f z ) 2 f Γ ( 1 2 ( h + j + 2 ) , ( - c - b - 2 f z ) 2 4 f ) ) + ( b - c ) 2 4 f j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b - c ) - h - j + 2 n ( - c + b - 2 f z ) h + j ( ( - c + b - 2 f z ) 2 f ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b - c ) ( - c + b - 2 f z ) Γ ( 1 2 ( h + j + 1 ) , ( - c + b - 2 f z ) 2 4 f ) - 2 f ( - c + b - 2 f z ) 2 f Γ ( 1 2 ( h + j + 2 ) , ( - c + b - 2 f z ) 2 4 f ) ) + - ( c - b ) 2 4 f j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c - b ) - h - j + 2 n ( c - b + 2 f z ) h + j ( - ( c - b + 2 f z ) 2 f ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c - b ) ( c - b + 2 f z ) Γ ( 1 2 ( h + j + 1 ) , - ( c - b + 2 f z ) 2 4 f ) + 2 - ( c - b + 2 f z ) 2 f f Γ ( 1 2 ( h + j + 2 ) , - ( c - b + 2 f z ) 2 4 f ) ) + - ( c + b ) 2 4 f j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c + b ) - h - j + 2 n ( c + b + 2 f z ) h + j ( - ( c + b + 2 f z ) 2 f ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( c + b ) ( c + b + 2 f z ) Γ ( 1 2 ( h + j + 1 ) , - ( c + b + 2 f z ) 2 4 f ) + 2 - ( c + b + 2 f z ) 2 f f Γ ( 1 2 ( h + j + 2 ) , - ( c + b + 2 f z ) 2 4 f ) ) ) /; n Condition z z n z 1 2 b z 1 2 c f z 2 -2 n -3 f -2 n 1 -1 c -1 b 2 4 f -1 h 0 j j 0 n -1 j -1 h 4 j -1 c -1 b -1 h -1 j 2 n -1 c -1 b -1 2 f z 1 2 h j -1 c -1 b -1 2 f z 1 2 2 f -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 c -1 b -1 c -1 b -1 2 f z 1 2 Gamma 1 2 h j 1 -1 c -1 b -1 2 f z 1 2 2 4 f -1 -1 2 f -1 c -1 b -1 2 f z 1 2 2 f -1 1 2 Gamma 1 2 h j 2 -1 c -1 b -1 2 f z 1 2 2 4 f -1 b -1 c 2 4 f -1 h 0 j j 0 n -1 j -1 h 4 j b -1 c -1 h -1 j 2 n -1 c b -1 2 f z 1 2 h j -1 c b -1 2 f z 1 2 2 f -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b -1 c -1 c b -1 2 f z 1 2 Gamma 1 2 h j 1 -1 c b -1 2 f z 1 2 2 4 f -1 -1 2 f -1 c b -1 2 f z 1 2 2 f -1 1 2 Gamma 1 2 h j 2 -1 c b -1 2 f z 1 2 2 4 f -1 -1 c -1 b 2 4 f -1 h 0 j j 0 n -1 j -1 h 4 j c -1 b -1 h -1 j 2 n c -1 b 2 f z 1 2 h j -1 c -1 b 2 f z 1 2 2 f -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c -1 b c -1 b 2 f z 1 2 Gamma 1 2 h j 1 -1 c -1 b 2 f z 1 2 2 4 f -1 2 -1 c -1 b 2 f z 1 2 2 f -1 1 2 f Gamma 1 2 h j 2 -1 c -1 b 2 f z 1 2 2 4 f -1 -1 c b 2 4 f -1 h 0 j j 0 n -1 j -1 h 4 j c b -1 h -1 j 2 n c b 2 f z 1 2 h j -1 c b 2 f z 1 2 2 f -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c b c b 2 f z 1 2 Gamma 1 2 h j 1 -1 c b 2 f z 1 2 2 4 f -1 2 -1 c b 2 f z 1 2 2 f -1 1 2 f Gamma 1 2 h j 2 -1 c b 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_"], " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["f", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "-", RowBox[List["2", " ", "f", " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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", "+", "j"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", "c", "-", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "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", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "j"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "j"]]], " ", SuperscriptBox["4", "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], "2"], "f"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "j"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", "c", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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[RowBox[List["\[ImaginaryI]", " ", "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", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "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