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/01.20.21.4166.01

 Input Form

 Integrate[Sinh[b z]^m Cosh[c Sqrt[z]]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) + (2^(-m - v)/b) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k E^(b (2 k - m) z) (-(-1)^m + E^(2 (-2 k + m) b z)) Binomial[m, k])/(-2 k + m), {k, 0, Floor[(1/2) (-1 + m)]}] + (1/c^2) (I^m 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(-2 s + v)^2) (E^((-2 s - v) c Sqrt[z]) (E^(4 s c Sqrt[z]) (-1 + c (2 s - v) Sqrt[z]) + E^(2 v c Sqrt[z]) (-1 + c (-2 s + v) Sqrt[z])) Binomial[v, s]), {s, 0, Floor[(1/2) (-1 + v)]}]) + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((2 (-1)^m E^(c (-2 s + v) Sqrt[z] + b (2 k - m) z))/(b (2 k - m)) + (2 E^((-c) (-2 s + v) Sqrt[z] + b (-2 k + m) z))/(b (-2 k + m)) - ((-1)^m c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 b (2 k - m) Sqrt[z])/(2 Sqrt[b (2 k - m)])])/ (E^((c^2 (-2 s + v)^2)/(4 b (2 k - m))) (b (2 k - m))^(3/2)) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 b (2 k - m) Sqrt[z])/ (2 Sqrt[b (-2 k + m)])])/(E^((c^2 (-2 s + v)^2)/ (4 b (-2 k + m))) (b (-2 k + m))^(3/2)) + (-1)^m ((2 E^((-c) (-2 s + v) Sqrt[z] + b (2 k - m) z))/ (b (2 k - m)) + (2 (-1)^m E^(c (-2 s + v) Sqrt[z] + b (-2 k + m) z))/(b (-2 k + m)) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[b (2 k - m)])])/(E^((c^2 (-2 s + v)^2)/(4 b (2 k - m))) (b (2 k - m))^(3/2)) - ((-1)^m c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 b (-2 k + m) Sqrt[z])/(2 Sqrt[b (-2 k + m)])])/(E^((c^2 (-2 s + v)^2)/(4 b (-2 k + m))) (b (-2 k + m))^(3/2)))), {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

 sinh m ( b z ) cosh v ( c z ) z m 2 - m - v z ( 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 ) + 2 - m - v ( 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 ) b k = 0 m - 1 2 ( - 1 ) k b ( 2 k - m ) z ( - ( - 1 ) m + 2 ( m - 2 k ) b z ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] m - 2 k + m 2 - m - v + 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 ) c 2 s = 0 v - 1 2 ( - 2 s - v ) c z ( 4 s c z ( c ( 2 s - v ) z - 1 ) + 2 v c z ( c ( v - 2 s ) z - 1 ) ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v - 2 s ) 2 + 2 - m - v - 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]]]]] ( ( - 1 ) m ( - c - c 2 ( v - 2 s ) 2 4 b ( 2 k - m ) π ( v - 2 s ) erfi ( 2 b z ( m - 2 k ) + c ( v - 2 s ) 2 b ( 2 k - m ) ) ( b ( 2 k - m ) ) 3 / 2 + 2 b ( 2 k - m ) z - c ( v - 2 s ) z b ( 2 k - m ) + 2 ( - 1 ) m c z ( v - 2 s ) + b ( m - 2 k ) z b ( m - 2 k ) - ( - 1 ) m c - c 2 ( v - 2 s ) 2 4 b ( m - 2 k ) π ( v - 2 s ) erfi ( 2 b z ( m - 2 k ) + c ( v - 2 s ) 2 b ( m - 2 k ) ) ( b ( m - 2 k ) ) 3 / 2 ) + 2 ( - 1 ) m c z ( v - 2 s ) + b ( 2 k - m ) z b ( 2 k - m ) - ( - 1 ) m c - c 2 ( v - 2 s ) 2 4 b ( 2 k - m ) π ( v - 2 s ) erfi ( 2 b z ( 2 k - m ) + c ( v - 2 s ) 2 b ( 2 k - m ) ) ( b ( 2 k - m ) ) 3 / 2 + 2 b ( m - 2 k ) z - c ( v - 2 s ) z b ( m - 2 k ) - c - c 2 ( v - 2 s ) 2 4 b ( m - 2 k ) π ( v - 2 s ) erfi ( 2 b z ( 2 k - m ) + c ( v - 2 s ) 2 b ( m - 2 k ) ) ( b ( m - 2 k ) ) 3 / 2 ) /; m + v + Condition z b z m c z 1 2 v m 2 -1 m -1 v z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 2 -1 m -1 v Binomial v v 2 -1 1 -1 \$CellContext`v 2 b -1 k 0 m -1 2 -1 -1 k b 2 k -1 m z -1 -1 m 2 m -1 2 k b z Binomial m k m -1 2 k -1 m 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c 2 -1 s 0 v -1 2 -1 -2 s -1 v c z 1 2 4 s c z 1 2 c 2 s -1 v z 1 2 -1 2 v c z 1 2 c v -1 2 s z 1 2 -1 Binomial v s v -1 2 s 2 -1 2 -1 m -1 v -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 m -1 c -1 c 2 v -1 2 s 2 4 b 2 k -1 m -1 1 2 v -1 2 s Erfi 2 b z 1 2 m -1 2 k c v -1 2 s 2 b 2 k -1 m 1 2 -1 b 2 k -1 m 3 2 -1 2 b 2 k -1 m z -1 c v -1 2 s z 1 2 b 2 k -1 m -1 2 -1 m c z 1 2 v -1 2 s b m -1 2 k z b m -1 2 k -1 -1 -1 m c -1 c 2 v -1 2 s 2 4 b m -1 2 k -1 1 2 v -1 2 s Erfi 2 b z 1 2 m -1 2 k c v -1 2 s 2 b m -1 2 k 1 2 -1 b m -1 2 k 3 2 -1 2 -1 m c z 1 2 v -1 2 s b 2 k -1 m z b 2 k -1 m -1 -1 -1 m c -1 c 2 v -1 2 s 2 4 b 2 k -1 m -1 1 2 v -1 2 s Erfi 2 b z 1 2 2 k -1 m c v -1 2 s 2 b 2 k -1 m 1 2 -1 b 2 k -1 m 3 2 -1 2 b m -1 2 k z -1 c v -1 2 s z 1 2 b m -1 2 k -1 -1 c -1 c 2 v -1 2 s 2 4 b m -1 2 k -1 1 2 v -1 2 s Erfi 2 b z 1 2 2 k -1 m c v -1 2 s 2 b m -1 2 k 1 2 -1 b m -1 2 k 3 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18