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 Cosh

 http://functions.wolfram.com/01.20.21.1143.01

 Input Form

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

 Standard Form

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FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], " ", 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]", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"]]], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]]]], "/;", 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 cos m ( d z ) cosh ( z c + g ) z 2 - m - 2 n - 2 k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( d ( m - 2 k ) ) - 2 ( n + 1 ) ( c 2 4 d ( m - 2 k ) - g j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ) - h - j + 2 n ( 2 d ( m - 2 k ) z - c ) h + j ( ( 2 d ( m - 2 k ) z - c ) 2 d ( m - 2 k ) ) 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 d ( m - 2 k ) ( 2 d ( m - 2 k ) z - c ) 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , ( 2 d ( m - 2 k ) z - c ) 2 4 d ( m - 2 k ) ) - c ( 2 d ( m - 2 k ) z - c ) Γ ( 1 2 ( h + j + 1 ) , ( 2 d ( m - 2 k ) z - c ) 2 4 d ( m - 2 k ) ) ) + c 2 4 d ( m - 2 k ) + g j = 0 n h = 0 j ( - 1 ) j - h 4 j c - h - j + 2 n ( c + 2 d ( m - 2 k ) z ) h + j ( ( c + 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) ) 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 d ( m - 2 k ) z ) Γ ( 1 2 ( h + j + 1 ) , ( c + 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) + 2 d ( m - 2 k ) ( c + 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , ( c + 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) ) + - c 2 4 d ( m - 2 k ) - g j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - c ) - h - j + 2 n ( - c - 2 d ( m - 2 k ) z ) h + j ( - ( - c - 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) ) 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 d ( m - 2 k ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( - c - 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) - 2 d ( m - 2 k ) - ( - c - 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , - ( - c - 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) ) + g - c 2 4 d ( m - 2 k ) j = 0 n h = 0 j ( - 1 ) j - h 4 j c - h - j + 2 n ( c - 2 d ( m - 2 k ) z ) h + j ( - ( c - 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) ) 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 d ( m - 2 k ) z ) Γ ( 1 2 ( h + j + 1 ) , - ( c - 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) - 2 d ( m - 2 k ) - ( c - 2 d ( m - 2 k ) z ) 2 d ( m - 2 k ) Γ ( 1 2 ( h + j + 2 ) , - ( c - 2 d ( m - 2 k ) z ) 2 4 d ( m - 2 k ) ) ) ) - 2 - m c - 2 ( n + 1 ) ( 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]]]]] ( g Γ ( 2 ( n + 1 ) , - c z ) + - g Γ ( 2 ( n + 1 ) , c z ) ) ( 1 - m mod 2 \$CellContext`m 2 ) /; n m + Condition z z n d z m z 1 2 c g 2 -1 m -1 2 n -2 k 0 m -1 2 -1 Binomial m k d m -1 2 k -2 n 1 c 2 4 d m -1 2 k -1 -1 g h 0 j j 0 n -1 j -1 h 4 j -1 c -1 h -1 j 2 n 2 d m -1 2 k z 1 2 -1 c h j 2 d m -1 2 k z 1 2 -1 c 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 d m -1 2 k 2 d m -1 2 k z 1 2 -1 c 2 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 2 d m -1 2 k z 1 2 -1 c 2 4 d m -1 2 k -1 -1 c 2 d m -1 2 k z 1 2 -1 c Gamma 1 2 h j 1 2 d m -1 2 k z 1 2 -1 c 2 4 d m -1 2 k -1 c 2 4 d m -1 2 k -1 g h 0 j j 0 n -1 j -1 h 4 j c -1 h -1 j 2 n c 2 d m -1 2 k z 1 2 h j c 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c c 2 d m -1 2 k z 1 2 Gamma 1 2 h j 1 c 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 2 d m -1 2 k c 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 c 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 -1 c 2 4 d m -1 2 k -1 -1 g h 0 j j 0 n -1 j -1 h 4 j -1 c -1 h -1 j 2 n -1 c -1 2 d m -1 2 k z 1 2 h j -1 -1 c -1 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j -1 c -1 c -1 2 d m -1 2 k z 1 2 Gamma 1 2 h j 1 -1 -1 c -1 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 -1 2 d m -1 2 k -1 -1 c -1 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 -1 -1 c -1 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 g -1 c 2 4 d m -1 2 k -1 h 0 j j 0 n -1 j -1 h 4 j c -1 h -1 j 2 n c -1 2 d m -1 2 k z 1 2 h j -1 c -1 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j c c -1 2 d m -1 2 k z 1 2 Gamma 1 2 h j 1 -1 c -1 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 -1 2 d m -1 2 k -1 c -1 2 d m -1 2 k z 1 2 2 d m -1 2 k -1 1 2 Gamma 1 2 h j 2 -1 c -1 2 d m -1 2 k z 1 2 2 4 d m -1 2 k -1 -1 2 -1 m c -2 n 1 Binomial m m 2 -1 g Gamma 2 n 1 -1 c z 1 2 -1 g Gamma 2 n 1 c 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