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 Cosh

 http://functions.wolfram.com/01.20.21.1443.01

 Input Form

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

 Standard Form

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", RowBox[List["(", RowBox[List["1", "+", "h", "+", "j"]], ")"]]]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", "p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], " ", 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", "+", "p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 z n p z cos m ( b z ) cosh ( c z ) z 2 - m - 2 n - 2 s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( b ( m - 2 s ) ) - 2 n - 2 ( ( p - c ) 2 4 b ( m - 2 s ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p - c ) - h - j + 2 n ( - c + p + 2 b ( m - 2 s ) z ) h + j ( ( - c + p + 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p - c ) ( - c + p + 2 b ( m - 2 s ) z ) Γ ( 1 2 ( h + j + 1 ) , ( - c + p + 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) + 2 b ( m - 2 s ) ( - c + p + 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) Γ ( 1 2 ( h + j + 2 ) , ( - c + p + 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) ) + ( c + p ) 2 4 b ( m - 2 s ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c + p ) - h - j + 2 n ( c + p + 2 b ( m - 2 s ) z ) h + j ( ( c + p + 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( c + p ) ( c + p + 2 b ( m - 2 s ) z ) Γ ( 1 2 ( h + j + 1 ) , ( c + p + 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) + 2 b ( m - 2 s ) ( c + p + 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) Γ ( 1 2 ( h + j + 2 ) , ( c + p + 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) ) + - ( p - c ) 2 4 b ( m - 2 s ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( p - c ) - h - j + 2 n ( - c + p - 2 b ( m - 2 s ) z ) h + j ( - ( - c + p - 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( p - c ) ( - c + p - 2 b ( m - 2 s ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( - c + p - 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) - 2 b ( m - 2 s ) - ( - c + p - 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) Γ ( 1 2 ( h + j + 2 ) , - ( - c + p - 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) ) + - ( c + p ) 2 4 b ( m - 2 s ) j = 0 n h = 0 j ( - 1 ) j - h 4 j ( c + p ) - h - j + 2 n ( c + p - 2 b ( m - 2 s ) z ) h + j ( - ( c + p - 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["h", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( c + p ) ( c + p - 2 b ( m - 2 s ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( c + p - 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) - 2 b ( m - 2 s ) - ( c + p - 2 b ( m - 2 s ) z ) 2 b ( m - 2 s ) Γ ( 1 2 ( h + j + 2 ) , - ( c + p - 2 b ( m - 2 s ) z ) 2 4 b ( m - 2 s ) ) ) ) - 2 - m ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( Γ ( 2 ( n + 1 ) , ( c - p ) z ) ( c - p ) - 2 ( n + 1 ) + ( c + p ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - c - p ) z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) /; n m + Condition z z n p z 1 2 b z m c z 1 2 2 -1 m -1 2 n -2 s 0 m -1 2 -1 Binomial m s b m -1 2 s -2 n -2 p -1 c 2 4 b m -1 2 s -1 h 0 j j 0 n -1 j -1 h 4 j p -1 c -1 h -1 j 2 n -1 c p 2 b m -1 2 s z 1 2 h j -1 c p 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p -1 c -1 c p 2 b m -1 2 s z 1 2 Gamma 1 2 h j 1 -1 c p 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 2 b m -1 2 s -1 c p 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 Gamma 1 2 h j 2 -1 c p 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 c p 2 4 b m -1 2 s -1 h 0 j j 0 n -1 j -1 h 4 j c p -1 h -1 j 2 n c p 2 b m -1 2 s z 1 2 h j c p 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c p c p 2 b m -1 2 s z 1 2 Gamma 1 2 h j 1 c p 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 2 b m -1 2 s c p 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 Gamma 1 2 h j 2 c p 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 -1 p -1 c 2 4 b m -1 2 s -1 h 0 j j 0 n -1 j -1 h 4 j p -1 c -1 h -1 j 2 n -1 c p -1 2 b m -1 2 s z 1 2 h j -1 -1 c p -1 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j p -1 c -1 c p -1 2 b m -1 2 s z 1 2 Gamma 1 2 h j 1 -1 -1 c p -1 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 -1 2 b m -1 2 s -1 -1 c p -1 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 Gamma 1 2 h j 2 -1 -1 c p -1 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 -1 c p 2 4 b m -1 2 s -1 h 0 j j 0 n -1 j -1 h 4 j c p -1 h -1 j 2 n c p -1 2 b m -1 2 s z 1 2 h j -1 c p -1 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c p c p -1 2 b m -1 2 s z 1 2 Gamma 1 2 h j 1 -1 c p -1 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 -1 2 b m -1 2 s -1 c p -1 2 b m -1 2 s z 1 2 2 b m -1 2 s -1 1 2 Gamma 1 2 h j 2 -1 c p -1 2 b m -1 2 s z 1 2 2 4 b m -1 2 s -1 -1 2 -1 m Binomial m m 2 -1 Gamma 2 n 1 c -1 p z 1 2 c -1 p -2 n 1 c p -2 n 1 Gamma 2 n 1 -1 c -1 p z 1 2 1 -1 \$CellContext`m 2 n m SuperPlus [/itex]

 Rule Form

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

 2002-12-18