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/01.20.21.4554.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Sinh[b z^r + e]^m Cosh[c z^r]^v, z] == 2^(-m - v) z^\[Alpha] ((Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]))/(I^m \[Alpha]) + (1/r) (Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^k Binomial[m, k] ((E^(e (-2 k + m)) Gamma[\[Alpha]/r, b (2 k - m) z^r])/(b (2 k - m) z^r)^(\[Alpha]/r) + ((-1)^m E^(2 e k - e m) Gamma[\[Alpha]/r, b (-2 k + m) z^r])/ (b (-2 k + m) z^r)^(\[Alpha]/r)), {k, 0, Floor[(1/2) (-1 + m)]}]) + (1/r) ((Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (Gamma[\[Alpha]/r, c (2 s - v) z^r]/ (c (2 s - v) z^r)^(\[Alpha]/r) + Gamma[\[Alpha]/r, c (-2 s + v) z^r]/(c (-2 s + v) z^r)^(\[Alpha]/r)), {s, 0, Floor[(1/2) (-1 + v)]}])/I^m) - (1/r) Sum[(-1)^k Binomial[m, k] Sum[E^(2 e k - e m) Binomial[v, s] ((E^(-4 e k + 2 e m) Gamma[\[Alpha]/r, (2 b k - b m + 2 c s - c v) z^r])/((2 b k - b m + 2 c s - c v) z^r)^(\[Alpha]/r) + ((-1)^m Gamma[\[Alpha]/r, (-2 b k + b m + 2 c s - c v) z^r])/ ((-2 b k + b m + 2 c s - c v) z^r)^(\[Alpha]/r) + (E^(-4 e k + 2 e m) Gamma[\[Alpha]/r, (2 b k - b m - 2 c s + c v) z^r])/((2 b k - b m - 2 c s + c v) z^r)^(\[Alpha]/r) + ((-1)^m Gamma[\[Alpha]/r, (-2 b k + b m - 2 c s + c v) z^r])/ ((-2 b k + b m - 2 c s + c v) z^r)^(\[Alpha]/r)), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}]) /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z α - 1 sinh m ( b z r + e ) cosh v ( c z r ) z 2 - m - v z α ( - 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 ) α + 1 r ( 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]]]]] ( e ( m - 2 k ) Γ ( α r , b ( 2 k - m ) z r ) ( b ( 2 k - m ) z r ) - α r + ( - 1 ) m 2 e k - e m ( b ( m - 2 k ) z r ) - α r Γ ( α r , b ( m - 2 k ) z r ) ) + - m r ( 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 s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( α r , c ( 2 s - v ) z r ) ( c ( 2 s - v ) z r ) - α r + ( c ( v - 2 s ) z r ) - α r Γ ( α r , c ( v - 2 s ) z r ) ) - 1 r 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 2 e k - e m ( 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 ( α r , ( - 2 b k + b m - 2 c s + c v ) z r ) ( ( - 2 b k + b m - 2 c s + c v ) z r ) - α r + 2 e m - 4 e k ( ( 2 b k - b m - 2 c s + c v ) z r ) - α r Γ ( α r , ( 2 b k - b m - 2 c s + c v ) z r ) + ( - 1 ) m ( ( - 2 b k + b m + 2 c s - c v ) z r ) - α r Γ ( α r , ( - 2 b k + b m + 2 c s - c v ) z r ) + 2 e m - 4 e k ( ( 2 b k - b m + 2 c s - c v ) z r ) - α r Γ ( α r , ( 2 b k - b m + 2 c s - c v ) z r ) ) ) /; m + v + Condition z z α -1 b z r e m c z r v 2 -1 m -1 v z α -1 m Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 α -1 1 r -1 Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k Binomial m k e m -1 2 k Gamma α r -1 b 2 k -1 m z r b 2 k -1 m z r -1 α r -1 -1 m 2 e k -1 e m b m -1 2 k z r -1 α r -1 Gamma α r -1 b m -1 2 k z r -1 m r -1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma α r -1 c 2 s -1 v z r c 2 s -1 v z r -1 α r -1 c v -1 2 s z r -1 α r -1 Gamma α r -1 c v -1 2 s z r -1 1 r -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 2 e k -1 e m Binomial v s -1 m Gamma α r -1 -2 b k b m -1 2 c s c v z r -2 b k b m -1 2 c s c v z r -1 α r -1 2 e m -1 4 e k 2 b k -1 b m -1 2 c s c v z r -1 α r -1 Gamma α r -1 2 b k -1 b m -1 2 c s c v z r -1 m -2 b k b m 2 c s -1 c v z r -1 α r -1 Gamma α r -1 -2 b k b m 2 c s -1 c v z r 2 e m -1 4 e k 2 b k -1 b m 2 c s -1 c v z r -1 α r -1 Gamma α r -1 2 b k -1 b m 2 c s -1 c v z r m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18