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/01.20.21.1569.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["c", "2"]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", "2"], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["c", " ", SqrtBox["z"]]]]], ")"]]]], "-", RowBox[List["c", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", 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")"]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "c"]], " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", "c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]], "-", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " 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RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]], "-", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", "c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], "]"]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 sinh m ( b z ) cosh ( c z ) z ( 2 ) m - c z ( - z c + 2 c z ( c z - 1 ) - 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 + 2 - m - 2 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]]]]] ( ( - 1 ) m ( - c - c 2 4 b ( 2 k - m ) π erfi ( c + 2 b ( m - 2 k ) z 2 b ( 2 k - m ) ) ( b ( 2 k - m ) ) 3 / 2 + 2 b ( 2 k - m ) z - c z b ( 2 k - m ) + 2 ( - 1 ) m z c + b ( m - 2 k ) z b ( m - 2 k ) - ( - 1 ) m c - c 2 4 b ( m - 2 k ) π erfi ( c + 2 b ( m - 2 k ) z 2 b ( m - 2 k ) ) ( b ( m - 2 k ) ) 3 / 2 ) + 2 ( - 1 ) m z c + b ( 2 k - m ) z b ( 2 k - m ) - ( - 1 ) m c - c 2 4 b ( 2 k - m ) π erfi ( c + 2 b ( 2 k - m ) z 2 b ( 2 k - m ) ) ( b ( 2 k - m ) ) 3 / 2 + 2 b ( m - 2 k ) z - c z b ( m - 2 k ) - c - c 2 4 b ( m - 2 k ) π erfi ( c + 2 b ( 2 k - m ) z 2 b ( m - 2 k ) ) ( b ( m - 2 k ) ) 3 / 2 ) /; m + Condition z b z m c z 1 2 2 -1 m -1 c z 1 2 -1 z 1 2 c 2 c z 1 2 c z 1 2 -1 -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c 2 -1 2 -1 m -2 k 0 m -1 2 -1 -1 k Binomial m k -1 m -1 c -1 c 2 4 b 2 k -1 m -1 1 2 Erfi c 2 b m -1 2 k z 1 2 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 z 1 2 b 2 k -1 m -1 2 -1 m z 1 2 c b m -1 2 k z b m -1 2 k -1 -1 -1 m c -1 c 2 4 b m -1 2 k -1 1 2 Erfi c 2 b m -1 2 k z 1 2 2 b m -1 2 k 1 2 -1 b m -1 2 k 3 2 -1 2 -1 m z 1 2 c b 2 k -1 m z b 2 k -1 m -1 -1 -1 m c -1 c 2 4 b 2 k -1 m -1 1 2 Erfi c 2 b 2 k -1 m z 1 2 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 z 1 2 b m -1 2 k -1 -1 c -1 c 2 4 b m -1 2 k -1 1 2 Erfi c 2 b 2 k -1 m z 1 2 2 b m -1 2 k 1 2 -1 b m -1 2 k 3 2 -1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List["b_", " ", "z_"]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", "2"], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "c"]], " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", SqrtBox["z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["c", " ", SqrtBox["z"]]]]], ")"]]]], "-", RowBox[List["c", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18