Gaussian Integrals

[Pages:1]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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