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/01.20.21.4017.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18