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 Exp

 http://functions.wolfram.com/01.03.21.0589.01

 Input Form

 Integrate[z^(2 n) a^(b z^2 + e) h^(c z^2 + g), z] == (-(1/2)) a^e h^g z^(1 + 2 n) ((-z^2) (b Log[a] + c Log[h]))^(-(1/2) - n) (Erfc[Sqrt[(-z^2) (b Log[a] + c Log[h])]] Gamma[1/2 + n] + a^(b z^2) h^(c z^2) Sum[((-z^2) (b Log[a] + c Log[h]))^(1/2 + j)/ Pochhammer[1/2 + n, 1 + j - n], {j, 0, -1 + n}] - a^(b z^2) h^(c z^2) Sum[((-z^2) (b Log[a] + c Log[h]))^(1/2 + j)/ Pochhammer[1/2 + n, 1 + j - n], {j, n, -1}]) /; Element[n, Integers]

 Standard Form

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 MathML Form

 z 2 n a b z 2 + e h c z 2 + g z - 1 2 a e h g z 2 n + 1 ( - z 2 ( b log ( a ) + c log ( h ) ) ) - n - 1 2 ( h c z 2 a b z 2 j = 0 n - 1 ( - z 2 ( b log ( a ) + c log ( h ) ) ) j + 1 2 ( n + 1 2 ) j - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "n", "+", "1"]]], Pochhammer] - h c z 2 a b z 2 j = n - 1 ( - z 2 ( b log ( a ) + c log ( h ) ) ) j + 1 2 ( n + 1 2 ) j - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "n", "+", "1"]]], Pochhammer] + erfc ( - z 2 ( b log ( a ) + c log ( h ) ) ) Γ ( n + 1 2 ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z 2 n a b z 2 e h c z 2 g -1 1 2 a e h g z 2 n 1 -1 z 2 b a c h -1 n -1 1 2 h c z 2 a b z 2 j 0 n -1 -1 z 2 b a c h j 1 2 Pochhammer n 1 2 j -1 n 1 -1 -1 h c z 2 a b z 2 j n -1 -1 z 2 b a c h j 1 2 Pochhammer n 1 2 j -1 n 1 -1 Erfc -1 z 2 b a c h 1 2 Gamma n 1 2 n [/itex]

 Rule Form

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

 2002-12-18