Gaussian Integrals
Z∞
−∞
e−x2dx =√π(1)
Z∞
0
e−ax2dx =1
2rπ
a(2)
Z∞
−∞
e−ax2+bx dx =eb2
4arπ
a(3)
Z∞
0
eiax2dx =1
2riπ
a(4)
Z∞
0
e−iax2dx =1
2rπ
ia(5)
In general, from dimensional anlysis we see:
Z∞
0
xne−ax2dx ∝a−(n+1
2)(6)
and in particular:
Z∞
0
xne−ax2dx =
(n−1)·(n−3)...3·1
2
n
2+1a
n
2pπ
a,for n even
[1
2(n−1)]!
2a
n+1
2
,for n odd (7)
Notes on proving these integrals: Integral 1 is done by squaring the integral, combining
the exponents to x2+y2switching to polar coordinates, and taking the R integral in the
limit as R→ ∞. Integral 2 is done by changing variables then using Integral 1. Integral 3 is
done by completing the square in the exponent and then changing variables to use equation 1.
Integral 4(5) can be done by integrating over a wedge with angle π
4(−π
4), using Cauchy’s
theory to relate the integral over the real number to the other side of the wedge, and then
using Integral 1.
For n even Integral 7 can be done by taking derivatives of equation 2 with respect to a.
For n odd, Integral 7 can be done with the substitution u=ax2, and then integrating by
parts.
