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/01.20.21.4184.01

 Input Form

 Integrate[Sinh[b z^2]^m Cosh[c z^2 + g]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) - (1/b) (2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/(-2 k + m)) ((-1)^k Binomial[m, k] ((-1)^m Sqrt[b (2 k - m)] Erfi[Sqrt[b (2 k - m)] z] + Sqrt[b (-2 k + m)] Erfi[(b (2 k - m) z)/Sqrt[b (-2 k + m)]])), {k, 0, Floor[(1/2) (-1 + m)]}]) - (1/c) (I^m 2^(-1 - m - v) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(-2 s + v)) (E^(g (2 s - v)) Binomial[v, s] (Sqrt[c (2 s - v)] Erfi[Sqrt[c (2 s - v)] z] + E^(2 g (-2 s + v)) Sqrt[c (-2 s + v)] Erfi[(c (2 s - v) z)/Sqrt[c (-2 s + v)]])), {s, 0, Floor[(1/2) (-1 + v)]}]) + 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((E^(g (-2 s + v)) Erfi[(-4 b k z + 2 b m z - 2 c (2 s - v) z)/ (2 Sqrt[b (-2 k + m) + c (-2 s + v)])])/ Sqrt[b (-2 k + m) + c (-2 s + v)] - ((-1)^m E^(g (2 s - v)) Sqrt[2 b k - b m + 2 c s - c v] Erfi[(4 b k z - 2 b m z + 2 c (2 s - v) z)/(2 Sqrt[2 b k - b m + 2 c s - c v])])/ (b (-2 k + m) + c (-2 s + v)) + (1/(-2 b k + b m + 2 c s - c v)) (E^(g (-2 s + v)) ((-(-1)^m) Sqrt[2 b k - b m - 2 c s + c v] Erfi[(4 b k z - 2 b m z - 2 c (2 s - v) z)/(2 Sqrt[2 b k - b m - 2 c s + c v])] + E^(2 g (2 s - v)) Sqrt[-2 b k + b m + 2 c s - c v] Erfi[(-4 b k z + 2 b m z + 2 c (2 s - v) z)/(2 Sqrt[-2 b k + b m + 2 c s - c v])]))), {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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RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "g", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b", " ", "k", " ", "z"]], "+", RowBox[List["2", " ", "b", " ", "m", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18