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/01.20.21.1300.01

 Input Form

 Integrate[E^(p z) Cos[b Sqrt[z]]^m Cosh[c Sqrt[z]], z] == 2^(-2 - m) Binomial[m, m/2] ((4 E^(p z) Cosh[c Sqrt[z]])/p + (c Sqrt[Pi] Erfi[(-c + 2 p Sqrt[z])/(2 Sqrt[p])])/ (E^(c^2/(4 p)) p^(3/2)) - (c Sqrt[Pi] Erfi[(c + 2 p Sqrt[z])/(2 Sqrt[p])])/ (E^(c^2/(4 p)) p^(3/2))) (1 - Mod[m, 2]) + 2^(-2 - m) Sum[Binomial[m, s] ((8 E^(p z) Cos[b (m - 2 s) Sqrt[z]] Cosh[c Sqrt[z]])/p - (1/p^(3/2)) (E^((I c + b (m - 2 s))^2/(4 p)) Sqrt[Pi] (I c + b (m - 2 s)) Erf[((-I) c - b (m - 2 s) - 2 I p Sqrt[z])/(2 Sqrt[p])]) + (1/p^(3/2)) (E^((I c - b m + 2 b s)^2/(4 p)) Sqrt[Pi] ((-I) c + b m - 2 b s) Erf[((-I) c + b (m - 2 s) - 2 I p Sqrt[z])/(2 Sqrt[p])]) + (1/p^(3/2)) (E^(((-I) c + b m - 2 b s)^2/(4 p)) Sqrt[Pi] ((-I) c + b m - 2 b s) Erf[((-I) c + b (m - 2 s) + 2 I p Sqrt[z])/ (2 Sqrt[p])]) - (1/p^(3/2)) (I E^((I c + b (m - 2 s))^2/(4 p)) Sqrt[Pi] (I c + b (m - 2 s)) Erfi[(-c + I b (m - 2 s) + 2 p Sqrt[z])/ (2 Sqrt[p])])), {s, 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["\[ExponentialE]", RowBox[List["p", " ", "z"]]], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]]]], "p"], "+", FractionBox[RowBox[List["c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["c", "2"], RowBox[List["4", " ", "p"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", 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FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "+", RowBox[List["2", " ", "p", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 p z cos m ( b z ) cosh ( c z ) z 2 - m - 2 ( 4 p z cosh ( c z ) p + c - c 2 4 p π erfi ( 2 p z - c 2 p ) p 3 / 2 - c - c 2 4 p π erfi ( c + 2 p z 2 p ) p 3 / 2 ) ( 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 ) + 2 - m - 2 s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 8 p z cos ( b ( m - 2 s ) z ) cosh ( c z ) p + ( - c + b m - 2 b s ) 2 4 p π ( - c + b m - 2 b s ) erf ( - c + b ( m - 2 s ) + 2 p z 2 p ) p 3 / 2 + ( c - b m + 2 b s ) 2 4 p π ( - c + b m - 2 b s ) erf ( - c + b ( m - 2 s ) - 2 p z 2 p ) p 3 / 2 - ( c + b ( m - 2 s ) ) 2 4 p π ( c + b ( m - 2 s ) ) erf ( - c - b ( m - 2 s ) - 2 p z 2 p ) p 3 / 2 - ( c + b ( m - 2 s ) ) 2 4 p π ( c + b ( m - 2 s ) ) erfi ( - c + b ( m - 2 s ) + 2 p z 2 p ) p 3 / 2 ) /; m + Condition z p z b z 1 2 m c z 1 2 2 -1 m -2 4 p z c z 1 2 p -1 c -1 c 2 4 p -1 1 2 Erfi 2 p z 1 2 -1 c 2 p 1 2 -1 p 3 2 -1 -1 c -1 c 2 4 p -1 1 2 Erfi c 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 2 -1 m -2 s 0 m -1 2 -1 Binomial m s 8 p z b m -1 2 s z 1 2 c z 1 2 p -1 -1 c b m -1 2 b s 2 4 p -1 1 2 -1 c b m -1 2 b s Erf -1 c b m -1 2 s 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 c -1 b m 2 b s 2 4 p -1 1 2 -1 c b m -1 2 b s Erf -1 c b m -1 2 s -1 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 -1 c b m -1 2 s 2 4 p -1 1 2 c b m -1 2 s Erf -1 c -1 b m -1 2 s -1 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 -1 c b m -1 2 s 2 4 p -1 1 2 c b m -1 2 s Erfi -1 c b m -1 2 s 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18