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; }

 Exp

 http://functions.wolfram.com/01.03.21.0598.01

 Input Form

 Integrate[z^n a^(b Sqrt[z]) h^(c Sqrt[z] + f z), z] == (2^(-1 - 2 n) Sum[(-1)^(-j + k) 4^k Binomial[k, j] Binomial[n, k] (b Log[a] + c Log[h])^(-j - k + 2 n) (b Log[a] + c Log[h] + 2 f Sqrt[z] Log[h])^(j + k) (-((b Log[a] + c Log[h] + 2 f Sqrt[z] Log[h])^2/(f Log[h])))^ ((1/2) (-1 - j - k)) (Gamma[(1/2) (1 + j + k), -((b Log[a] + c Log[h] + 2 f Sqrt[z] Log[h])^2/(4 f Log[h]))] (b Log[a] + c Log[h]) (b Log[a] + c Log[h] + 2 f Sqrt[z] Log[h]) + 2 f Gamma[(1/2) (2 + j + k), -((b Log[a] + c Log[h] + 2 f Sqrt[z] Log[h])^2/(4 f Log[h]))] Log[h] Sqrt[-((b Log[a] + c Log[h] + 2 f Sqrt[z] Log[h])^2/(f Log[h]))]), {k, 0, n}, {j, 0, k}])/(E^((b Log[a] + c Log[h])^2/(4 f Log[h])) (f Log[h])^(2 (1 + n))) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["a", RowBox[List["b", " ", SqrtBox["z"]]]], SuperscriptBox["h", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "-", "k"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "j", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "j", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]], "]"]], " ", RowBox[List["Log", "[", "h", "]"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n a b z h z c + f z z 2 - 2 n - 1 - ( b log ( a ) + c log ( h ) ) 2 4 f log ( h ) ( f log ( h ) ) - 2 ( n + 1 ) k = 0 n j = 0 k ( - 1 ) k - j 4 k ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( b log ( a ) + c log ( h ) ) - j - k + 2 n ( b log ( a ) + c log ( h ) + 2 f z log ( h ) ) j + k ( - ( b log ( a ) + c log ( h ) + 2 f z log ( h ) ) 2 f log ( h ) ) 1 2 ( - j - k - 1 ) ( 2 f Γ ( 1 2 ( j + k + 2 ) , - ( b log ( a ) + c log ( h ) + 2 f z log ( h ) ) 2 4 f log ( h ) ) - ( b log ( a ) + c log ( h ) + 2 f z log ( h ) ) 2 f log ( h ) log ( h ) + Γ ( 1 2 ( j + k + 1 ) , - ( b log ( a ) + c log ( h ) + 2 f z log ( h ) ) 2 4 f log ( h ) ) ( b log ( a ) + c log ( h ) ) ( b log ( a ) + c log ( h ) + 2 f z log ( h ) ) ) /; n Condition z z n a b z 1 2 h z 1 2 c f z 2 -2 n -1 -1 b a c h 2 4 f h -1 f h -2 n 1 j 0 k k 0 n -1 k -1 j 4 k Binomial k j Binomial n k b a c h -1 j -1 k 2 n b a c h 2 f z 1 2 h j k -1 b a c h 2 f z 1 2 h 2 f h -1 1 2 -1 j -1 k -1 2 f Gamma 1 2 j k 2 -1 b a c h 2 f z 1 2 h 2 4 f h -1 -1 b a c h 2 f z 1 2 h 2 f h -1 1 2 h Gamma 1 2 j k 1 -1 b a c h 2 f z 1 2 h 2 4 f h -1 b a c h b a c h 2 f z 1 2 h n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["a_", RowBox[List["b_", " ", SqrtBox["z_"]]]], " ", SuperscriptBox["h_", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "-", "k"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "j", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", "f", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "j", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]], "]"]], " ", RowBox[List["Log", "[", "h", "]"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["f", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18