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/01.20.21.4552.01

 Input Form

 Integrate[z^n Sinh[b z^2]^m Cosh[c z^2]^v, z] == 2^(-1 - m - v) z^(1 + n) ((2 Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]))/(I^m (1 + n)) + Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^k Binomial[m, k] ((b (2 k - m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, b (2 k - m) z^2] + (-1)^m (b (-2 k + m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, b (-2 k + m) z^2]), {k, 0, Floor[(1/2) (-1 + m)]}] + (Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] ((c (2 s - v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, c (2 s - v) z^2] + (c (-2 s + v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, c (-2 s + v) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m - Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (((2 b k - b m + 2 c s - c v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (2 b k - b m + 2 c s - c v) z^2] + (-1)^m ((-2 b k + b m + 2 c s - c v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/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) (-1 - n)) Gamma[(1 + n)/2, (2 b k - b m - 2 c s + c v) z^2] + (-1)^m ((-2 b k + b m - 2 c s + c v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-2 b k + 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 sinh m ( b z 2 ) cosh v ( c z 2 ) z 2 - m - v - 1 z n + 1 ( 2 - m ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( n + 1 2 , b ( 2 k - m ) z 2 ) ( b ( 2 k - m ) z 2 ) 1 2 ( - n - 1 ) + ( - 1 ) m ( b ( m - 2 k ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , b ( m - 2 k ) z 2 ) ) + ( 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 ( m mod 2 \$CellContext`m 2 - 1 ) 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]]]]] ( Γ ( n + 1 2 , c ( 2 s - v ) z 2 ) ( c ( 2 s - v ) z 2 ) 1 2 ( - n - 1 ) + ( c ( v - 2 s ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , c ( v - 2 s ) z 2 ) ) - k = 0 m - 1 2 ( - 1 ) k ( 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]]]]] ( ( - 1 ) 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 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 ) + ( - 1 ) 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 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 m c z 2 v 2 -1 m -1 v -1 z n 1 2 -1 m Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k Binomial m k Gamma n 1 2 -1 b 2 k -1 m z 2 b 2 k -1 m z 2 1 2 -1 n -1 -1 m b m -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 b m -1 2 k z 2 Binomial m m 2 -1 -1 m \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma n 1 2 -1 c 2 s -1 v z 2 c 2 s -1 v z 2 1 2 -1 n -1 c v -1 2 s z 2 1 2 -1 n -1 Gamma n 1 2 -1 c v -1 2 s z 2 -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 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 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 -1 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 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