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/01.20.21.4074.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]], "+", "e"]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "d"], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "d"]]]], " ", 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RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["d"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["d", " ", "z"]]]], "]"]]]], "d"], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", 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 Date Added to functions.wolfram.com (modification date)

 2002-12-18