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/01.20.21.4556.01

 Input Form

 Integrate[z^n Sinh[b Sqrt[z] + e]^m Cosh[c Sqrt[z]]^v, z] == 2^(-m - v) ((1/(1 + n)) ((z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]))/I^m) + 2 b^(-2 - 2 n) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[((-1)^k E^(2 e k - e m) Binomial[m, k] (E^(-4 e k + 2 e m) Gamma[2 (1 + n), b (2 k - m) Sqrt[z]] + (-1)^m Gamma[2 (1 + n), b (-2 k + m) Sqrt[z]]))/ (-2 k + m)^(2 (1 + n)), {k, 0, Floor[(1/2) (-1 + m)]}] + (2 c^(-2 - 2 n) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(Binomial[v, s] (Gamma[2 (1 + n), c (2 s - v) Sqrt[z]] + Gamma[2 (1 + n), c (-2 s + v) Sqrt[z]]))/(-2 s + v)^(2 (1 + n)), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m - 2 z^(1 + n) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((E^(e (-2 k + m)) Gamma[2 (1 + n), (2 b k - b m + 2 c s - c v) Sqrt[z]])/ ((2 b k - b m + 2 c s - c v) Sqrt[z])^(2 (1 + n)) + ((-1)^m E^(2 e k - e m) Gamma[2 (1 + n), (-2 b k + b m + 2 c s - c v) Sqrt[z]])/((-2 b k + b m + 2 c s - c v) Sqrt[z])^ (2 (1 + n)) + (E^(-2 e k + e m) Gamma[2 (1 + n), (2 b k - b m - 2 c s + c v) Sqrt[z]])/ ((2 b k - b m - 2 c s + c v) Sqrt[z])^(2 (1 + n)) + ((-1)^m E^(2 e k - e m) Gamma[2 (1 + n), (b (-2 k + m) + c (-2 s + v)) Sqrt[z]])/((b (-2 k + m) + c (-2 s + v)) Sqrt[z])^ (2 (1 + n))), {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 ( z b + e ) cosh v ( c z ) z 2 - m - v ( z n + 1 - 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 + 2 b - 2 n - 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) k = 0 m - 1 2 ( - 1 ) k 2 e k - e m ( m - 2 k ) - 2 ( n + 1 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 e m - 4 e k Γ ( 2 ( n + 1 ) , b ( 2 k - m ) z ) + ( - 1 ) m Γ ( 2 ( n + 1 ) , b ( m - 2 k ) z ) ) + 2 - m c - 2 n - 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 mod 2 \$CellContext`m 2 - 1 ) s = 0 v - 1 2 ( v - 2 s ) - 2 ( n + 1 ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( 2 ( n + 1 ) , c ( 2 s - v ) z ) + Γ ( 2 ( n + 1 ) , c ( v - 2 s ) z ) ) - 2 z n + 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]]]]] 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]]]]] ( e m - 2 e k Γ ( 2 ( n + 1 ) , ( 2 b k - b m - 2 c s + c v ) z ) ( ( 2 b k - b m - 2 c s + c v ) z ) - 2 ( n + 1 ) + ( - 1 ) m 2 e k - e m ( ( - 2 b k + b m + 2 c s - c v ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( - 2 b k + b m + 2 c s - c v ) z ) + e ( m - 2 k ) ( ( 2 b k - b m + 2 c s - c v ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( 2 b k - b m + 2 c s - c v ) z ) + ( - 1 ) m 2 e k - e m ( ( b ( m - 2 k ) + c ( v - 2 s ) ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( b ( m - 2 k ) + c ( v - 2 s ) ) z ) ) ) /; n m + v + Condition z z n z 1 2 b e m c z 1 2 v 2 -1 m -1 v z n 1 -1 m Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 2 b -2 n -2 Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k 2 e k -1 e m m -1 2 k -2 n 1 Binomial m k 2 e m -1 4 e k Gamma 2 n 1 b 2 k -1 m z 1 2 -1 m Gamma 2 n 1 b m -1 2 k z 1 2 2 -1 m c -2 n -2 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 v -1 2 s -2 n 1 Binomial v s Gamma 2 n 1 c 2 s -1 v z 1 2 Gamma 2 n 1 c v -1 2 s z 1 2 -1 2 z n 1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s e m -1 2 e k Gamma 2 n 1 2 b k -1 b m -1 2 c s c v z 1 2 2 b k -1 b m -1 2 c s c v z 1 2 -2 n 1 -1 m 2 e k -1 e m -2 b k b m 2 c s -1 c v z 1 2 -2 n 1 Gamma 2 n 1 -2 b k b m 2 c s -1 c v z 1 2 e m -1 2 k 2 b k -1 b m 2 c s -1 c v z 1 2 -2 n 1 Gamma 2 n 1 2 b k -1 b m 2 c s -1 c v z 1 2 -1 m 2 e k -1 e m b m -1 2 k c v -1 2 s z 1 2 -2 n 1 Gamma 2 n 1 b m -1 2 k c v -1 2 s z 1 2 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18