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 Cosh

 http://functions.wolfram.com/01.20.21.1435.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2002-12-18