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/01.20.21.4167.01

 Input Form

 Integrate[Sinh[d z + e]^m Cosh[c z^2]^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)/d) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k E^((-2 k + m) (e + d z)) (-1 + (-1)^m E^(2 (2 k - m) (e + d z))) Binomial[m, k])/(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)) Binomial[v, s] (Sqrt[c (2 s - v)] Erfi[Sqrt[c (2 s - v)] z] + 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^(e (-2 k + m) + (-2 d k + d m)^2/(8 c s - 4 c v)) Erfi[(-2 d k + d m - 2 c (2 s - v) z)/(2 Sqrt[c (-2 s + v)])])/ Sqrt[c (-2 s + v)] - (1/(c (-2 s + v))) ((-1)^m E^(e (2 k - m) - (-2 d k + d m)^2/(8 c s - 4 c v)) Sqrt[2 c s - c v] Erfi[(2 d k - d m + 2 c (2 s - v) z)/ (2 Sqrt[2 c s - c v])]) + (1/(2 c s - c v)) (E^((-e) (2 k + m) - (2 d k - d m)^2/(-8 c s + 4 c v)) ((-(-1)^m) E^(4 e k) Sqrt[-2 c s + c v] Erfi[(2 d k - d m - 2 c (2 s - v) z)/(2 Sqrt[-2 c s + c v])] + E^(2 (e m + (2 d k - d m)^2/(-8 c s + 4 c v))) Sqrt[2 c s - c v] Erfi[(-2 d k + d m + 2 c (2 s - v) z)/(2 Sqrt[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["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["e", " ", "m"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", 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 MathML Form

 sinh m ( e + d z ) cosh v ( c z 2 ) 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 ) d k = 0 m - 1 2 ( - 1 ) k ( m - 2 k ) ( e + d z ) ( - 1 + ( - 1 ) m 2 ( 2 k - m ) ( e + 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 ( 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 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c ( v - 2 s ) erfi ( c ( 2 s - v ) z c ( v - 2 s ) ) + c ( 2 s - v ) erfi ( c ( 2 s - v ) z ) ) 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]]]]] ( ( d m - 2 d k ) 2 8 c s - 4 c v + e ( m - 2 k ) erfi ( - 2 d k + d m - 2 c ( 2 s - v ) z 2 c ( v - 2 s ) ) c ( v - 2 s ) + 1 2 c s - c v ( - ( 2 d k - d m ) 2 4 c v - 8 c s - e ( 2 k + m ) ( 2 ( ( 2 d k - d m ) 2 4 c v - 8 c s + e m ) 2 c s - c v erfi ( - 2 d k + d m + 2 c ( 2 s - v ) z 2 2 c s - c v ) - ( - 1 ) m 4 e k c v - 2 c s erfi ( 2 d k - d m - 2 c ( 2 s - v ) z 2 c v - 2 c s ) ) ) - ( - 1 ) m e ( 2 k - m ) - ( d m - 2 d k ) 2 8 c s - 4 c v 2 c s - c v erfi ( 2 d k - d m + 2 c ( 2 s - v ) z 2 2 c s - c v ) c ( v - 2 s ) ) /; m + v + Condition z e d z m c z 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 d -1 k 0 m -1 2 -1 -1 k m -1 2 k e d z -1 -1 m 2 2 k -1 m e d z Binomial m k 2 k -1 m -1 -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 Binomial v 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 v -1 2 s -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 d m -1 2 d k 2 8 c s -1 4 c v -1 e m -1 2 k Erfi -2 d k d m -1 2 c 2 s -1 v 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 2 4 c v -1 8 c s -1 -1 e 2 k m 2 2 d k -1 d m 2 4 c v -1 8 c s -1 e m 2 c s -1 c v 1 2 Erfi -2 d k d m 2 c 2 s -1 v z 2 2 c s -1 c v 1 2 -1 -1 -1 m 4 e k c v -1 2 c s 1 2 Erfi 2 d k -1 d m -1 2 c 2 s -1 v z 2 c v -1 2 c s 1 2 -1 -1 -1 m e 2 k -1 m -1 d m -1 2 d k 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 c 2 s -1 v 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