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/01.20.21.3682.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18