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 Exp

 http://functions.wolfram.com/01.03.21.0579.01

 Input Form

 Integrate[z^n a^(d z + e) h^(c z^2 + g), z] == ((-(1/2)) a^e h^g (c Log[h])^(-1 - n) Sum[2^(j - n) Binomial[n, j] Gamma[(1 + j)/2, -((d Log[a] + 2 c z Log[h])^2/(4 c Log[h]))] ((-d) Log[a])^(-j + n) (d Log[a] + 2 c z Log[h])^(1 + j) (-((d Log[a] + 2 c z Log[h])^2/(c Log[h])))^((1/2) (-1 - j)), {j, 0, n}])/E^((d^2 Log[a]^2)/(4 c Log[h])) /; 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[RowBox[List["d", " ", "z"]], "+", "e"]]], SuperscriptBox["h", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", "g"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["a", "e"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["d", "2"], " ", SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], "2"]]], RowBox[List["4", " ", "c", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]], " ", SuperscriptBox["h", "g"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List["j", "-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "j"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["2", " ", "c", " ", "z", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["4", " ", "c", " ", RowBox[List["Log", "[", "h", "]"]]]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], " ", RowBox[List["Log", "[", "a", "]"]]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["2", " ", "c", " ", "z", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], RowBox[List["1", "+", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", RowBox[List["Log", "[", "a", "]"]]]], "+", RowBox[List["2", " ", "c", " ", "z", " ", RowBox[List["Log", "[", "h", "]"]]]]]], ")"]], "2"], RowBox[List["c", " ", RowBox[List["Log", "[", "h", "]"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j"]], ")"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n a e + d z h c z 2 + g z - 1 2 a e - d 2 log 2 ( a ) 4 c log ( h ) h g ( c log ( h ) ) - n - 1 j = 0 n 2 j - n ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( j + 1 2 , - ( d log ( a ) + 2 c z log ( h ) ) 2 4 c log ( h ) ) ( - d log ( a ) ) n - j ( d log ( a ) + 2 c z log ( h ) ) j + 1 ( - ( d log ( a ) + 2 c z log ( h ) ) 2 c log ( h ) ) 1 2 ( - j - 1 ) /; n Condition z z n a e d z h c z 2 g -1 1 2 a e -1 d 2 a 2 4 c h -1 h g c h -1 n -1 j 0 n 2 j -1 n Binomial n j Gamma j 1 2 -1 -1 d a 2 c z h 2 4 c h -1 -1 d a n -1 j d a 2 c z h j 1 -1 d a 2 c z h 2 c h -1 1 2 -1 j -1 n [/itex]

 Rule Form

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

 2002-12-18