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/01.20.21.4193.01

 Input Form

 Integrate[Sinh[d z]^m Cosh[c z^2 + f 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]) - (1/d) (2^(-m - v) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(1/(2 k - m)) ((-1)^k E^((-2 k + m) d z) (-1 + (-1)^m E^(2 (2 k - m) d z)) Binomial[m, k]), {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^((f^2 (2 s - v))/(4 c)) Binomial[v, s] (E^((f^2 (-2 s + v))/(2 c)) Sqrt[c (2 s - v)] Erfi[(Sqrt[c (2 s - v)] (f + 2 c z))/(2 c)] + Sqrt[c (-2 s + v)] Erfi[((2 s - v) (f + 2 c z))/(2 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] ((1/Sqrt[c (-2 s + v)]) (E^((-2 d k + d m - 2 f s + f v)^2/ (8 c s - 4 c v)) Erfi[(-2 d k + d m - (2 s - v) (f + 2 c z))/ (2 Sqrt[c (-2 s + v)])]) - (1/(c (-2 s + v))) (((-1)^m Sqrt[2 c s - c v] Erfi[(2 d k - d m + (2 s - v) (f + 2 c z))/(2 Sqrt[2 c s - c v])])/ E^((-2 d k + d m - 2 f s + f v)^2/(8 c s - 4 c v))) + (1/(2 c s - c v)) (((-(-1)^m) Sqrt[-2 c s + c v] Erfi[(2 d k - d m - (2 s - v) (f + 2 c z))/(2 Sqrt[-2 c s + c v])] + E^((2 d k - d m - 2 f s + f v)^2/(-4 c s + 2 c v)) Sqrt[2 c s - c v] Erfi[(-2 d k + d m + (2 s - v) (f + 2 c z))/(2 Sqrt[2 c s - c v])])/E^((2 d k - d m + f (-2 s + v))^2/ (-8 c s + 4 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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"m"]], "+", RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]], "+", 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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 ( d z ) cosh v ( c z 2 + f 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) d k = 0 m - 1 2 ( - 1 ) k ( m - 2 k ) d z ( - 1 + ( - 1 ) m 2 ( 2 k - m ) d z ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 k - m - m 2 - m - v - 1 π c ( 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 ) s = 0 v - 1 2 1 v - 2 s ( f 2 ( 2 s - v ) 4 c ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( f 2 ( v - 2 s ) 2 c c ( 2 s - v ) erfi ( c ( 2 s - v ) ( f + 2 c z ) 2 c ) + c ( v - 2 s ) erfi ( ( 2 s - v ) ( f + 2 c z ) 2 c ( v - 2 s ) ) ) ) + 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]]]]] ( ( - 2 d k + d m - 2 f s + f v ) 2 8 c s - 4 c v erfi ( - 2 d k + d m - ( 2 s - v ) ( f + 2 c z ) 2 c ( v - 2 s ) ) c ( v - 2 s ) + 1 2 c s - c v ( - ( 2 d k - d m + f ( v - 2 s ) ) 2 4 c v - 8 c s ( ( 2 d k - d m - 2 f s + f v ) 2 2 c v - 4 c s 2 c s - c v erfi ( - 2 d k + d m + ( 2 s - v ) ( f + 2 c z ) 2 2 c s - c v ) - ( - 1 ) m c v - 2 c s erfi ( 2 d k - d m - ( 2 s - v ) ( f + 2 c z ) 2 c v - 2 c s ) ) ) - ( - 1 ) m - ( - 2 d k + d m - 2 f s + f v ) 2 8 c s - 4 c v 2 c s - c v erfi ( 2 d k - d m + ( 2 s - v ) ( f + 2 c z ) 2 2 c s - c v ) c ( v - 2 s ) ) /; m + v + Condition z d z m c z 2 f z 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 Binomial v v 2 -1 \$CellContext`v 2 -1 d -1 k 0 m -1 2 -1 -1 k m -1 2 k d z -1 -1 m 2 2 k -1 m d z Binomial m k 2 k -1 m -1 -1 m 2 -1 m -1 v -1 1 2 c -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 1 v -1 2 s -1 f 2 2 s -1 v 4 c -1 Binomial v s f 2 v -1 2 s 2 c -1 c 2 s -1 v 1 2 Erfi c 2 s -1 v 1 2 f 2 c z 2 c -1 c v -1 2 s 1 2 Erfi 2 s -1 v f 2 c z 2 c v -1 2 s 1 2 -1 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 -2 d k d m -1 2 f s f v 2 8 c s -1 4 c v -1 Erfi -2 d k d m -1 2 s -1 v f 2 c z 2 c v -1 2 s 1 2 -1 c v -1 2 s 1 2 -1 1 2 c s -1 c v -1 -1 2 d k -1 d m f v -1 2 s 2 4 c v -1 8 c s -1 2 d k -1 d m -1 2 f s f v 2 2 c v -1 4 c s -1 2 c s -1 c v 1 2 Erfi -2 d k d m 2 s -1 v f 2 c z 2 2 c s -1 c v 1 2 -1 -1 -1 m c v -1 2 c s 1 2 Erfi 2 d k -1 d m -1 2 s -1 v f 2 c z 2 c v -1 2 c s 1 2 -1 -1 -1 m -1 -2 d k d m -1 2 f s f v 2 8 c s -1 4 c v -1 2 c s -1 c v 1 2 Erfi 2 d k -1 d m 2 s -1 v f 2 c z 2 2 c s -1 c v 1 2 -1 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