Gaussian Integrals - University of Pennsylvania
Gaussian Integrals
e-x2
dx
=
(1)
-
e-ax2
dx
=
1
(2)
0
2a
e-ax2+bx dx
=
b2
e 4a
(3)
-
a
eiax2
dx
=
1
i
(4)
0
2a
e-iax2
dx
=
1
(5)
0
2 ia
In general, from dimensional anlysis we see:
xne-ax2 dx
a-(
n+1 2
)
(6)
0
and in particular:
0
xne-ax2
dx
=
(n-1)?(n-3)...3?1
2
n 2
+1
a
n 2
[
1 2
(n-1)]!
n+1
,
a
,
for n even
for n odd
(7)
2a 2
Notes on proving these integrals: Integral 1 is done by squaring the integral, combining the exponents to x2 + y2 switching 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
4
(-
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.
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