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/01.20.21.0903.01

 Input Form

 Integrate[z^n Sin[b z] Cosh[c Sqrt[z]], z] == (I (-1)^(-1 + n) 2^(-3 - 2 n) (Sum[(-1)^(-h + j) 4^j (-c)^(-h - j + 2 n) (-c - 2 I b Sqrt[z])^(h + j) (-((I (-c - 2 I b Sqrt[z])^2)/b))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-c - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (-c - 2 I b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (-c - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + j), -((I (-c - 2 I b Sqrt[z])^2)/(4 b))]), {j, 0, n}, {h, 0, j}]/ E^((I c^2)/(4 b)) + Sum[(-1)^(-h + j) 4^j c^(-h - j + 2 n) (c - 2 I b Sqrt[z])^(h + j) (-((I (c - 2 I b Sqrt[z])^2)/b))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (c - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I (c - 2 I b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (c - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + j), -((I (c - 2 I b Sqrt[z])^2)/(4 b))]), {j, 0, n}, {h, 0, j}]/ E^((I c^2)/(4 b)) - E^((I c^2)/(4 b)) Sum[(-1)^(-h + j) 4^j (-c)^(-h - j + 2 n) (-c + 2 I b Sqrt[z])^(h + j) ((I (-c + 2 I b Sqrt[z])^2)/b)^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-c) (-c + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (-c + 2 I b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (-c + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + j), (I (-c + 2 I b Sqrt[z])^2)/(4 b)]), {j, 0, n}, {h, 0, j}] - E^((I c^2)/(4 b)) Sum[(-1)^(-h + j) 4^j c^(-h - j + 2 n) (c + 2 I b Sqrt[z])^(h + j) ((I (c + 2 I b Sqrt[z])^2)/b)^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (c (c + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + j), (I (c + 2 I b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (c + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + j), (I (c + 2 I b Sqrt[z])^2)/(4 b)]), {j, 0, n}, {h, 0, j}]))/ b^(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["Sin", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", 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["-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]], ")"]], 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["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", 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["c", RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]], ")"]], 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["c", " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]], " ", 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["-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"], ")"]], 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["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]], " ", 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["c", RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"], ")"]], 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["c", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n sin ( b z ) cosh ( c z ) z ( - 1 ) n - 1 2 - 2 n - 3 b - 2 ( n + 1 ) ( - c 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ) - h - j + 2 n ( 2 b z - c ) h + j ( ( 2 b z - c ) 2 b ) 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]]]]] ( 2 b ( 2 b z - c ) 2 b Γ ( 1 2 ( h + j + 2 ) , ( 2 b z - c ) 2 4 b ) - c ( 2 b z - c ) Γ ( 1 2 ( h + j + 1 ) , ( 2 b z - c ) 2 4 b ) ) - c 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j c - h - j + 2 n ( c + 2 b z ) h + j ( ( c + 2 b z ) 2 b ) 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 ( c + 2 b z ) Γ ( 1 2 ( h + j + 1 ) , ( c + 2 b z ) 2 4 b ) + 2 ( c + 2 b z ) 2 b b Γ ( 1 2 ( h + j + 2 ) , ( c + 2 b z ) 2 4 b ) ) + - c 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ) - h - j + 2 n ( - c - 2 b z ) h + j ( - ( - c - 2 b z ) 2 b ) 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 ( - c - 2 b z ) Γ ( 1 2 ( h + j + 1 ) , - ( - c - 2 b z ) 2 4 b ) - 2 b - ( - c - 2 b z ) 2 b Γ ( 1 2 ( h + j + 2 ) , - ( - c - 2 b z ) 2 4 b ) ) + - c 2 4 b j = 0 n h = 0 j ( - 1 ) j - h 4 j c - h - j + 2 n ( c - 2 b z ) h + j ( - ( c - 2 b z ) 2 b ) 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 ( c - 2 b z ) Γ ( 1 2 ( h + j + 1 ) , - ( c - 2 b z ) 2 4 b ) - 2 b - ( c - 2 b z ) 2 b Γ ( 1 2 ( h + j + 2 ) , - ( c - 2 b z ) 2 4 b ) ) ) /; n Condition z z n b z c z 1 2 -1 n -1 2 -2 n -3 b -2 n 1 -1 c 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j -1 c -1 h -1 j 2 n 2 b z 1 2 -1 c h j 2 b z 1 2 -1 c 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 b 2 b z 1 2 -1 c 2 b -1 1 2 Gamma 1 2 h j 2 2 b z 1 2 -1 c 2 4 b -1 -1 c 2 b z 1 2 -1 c Gamma 1 2 h j 1 2 b z 1 2 -1 c 2 4 b -1 -1 c 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j c -1 h -1 j 2 n c 2 b z 1 2 h j c 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c c 2 b z 1 2 Gamma 1 2 h j 1 c 2 b z 1 2 2 4 b -1 2 c 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h j 2 c 2 b z 1 2 2 4 b -1 -1 c 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j -1 c -1 h -1 j 2 n -1 c -1 2 b z 1 2 h j -1 -1 c -1 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 c -1 c -1 2 b z 1 2 Gamma 1 2 h j 1 -1 -1 c -1 2 b z 1 2 2 4 b -1 -1 2 b -1 -1 c -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h j 2 -1 -1 c -1 2 b z 1 2 2 4 b -1 -1 c 2 4 b -1 h 0 j j 0 n -1 j -1 h 4 j c -1 h -1 j 2 n c -1 2 b z 1 2 h j -1 c -1 2 b z 1 2 2 b -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c c -1 2 b z 1 2 Gamma 1 2 h j 1 -1 c -1 2 b z 1 2 2 4 b -1 -1 2 b -1 c -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h j 2 -1 c -1 2 b z 1 2 2 4 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["b_", " ", "z_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", 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["-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]], ")"]], 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["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", 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["c", RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]], ")"]], 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["c", " ", RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]], " ", 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["-", "c"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"], ")"]], 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["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]]]], ")"]]]]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["c", "2"]]], RowBox[List["4", " ", "b"]]]], " ", 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["c", RowBox[List[RowBox[List["-", "h"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"], ")"]], 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["c", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], "b"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18