130_notes.dvi

This is just the extension of the Fourier series to allx.

Iff(x) is normalized, thenA(k) will also be normalized with this (symmetric) form of the Fourier
Transform. Thus, iff(x) is a probability amplitude in position-space,A(k) can be a probability
amplitude (in k-space).

5.6.3 Integral of Gaussian

This is just a slick derivation of the definite integral of a Gaussian from minus infinity to infinity.
With other limits, the integral cannot be done analytically but is tabulated. Functions are available
in computer libraries to return this important integral.

The answer is ∞
∫

−∞

dx e−ax

2

=

√

π

a

.

Define

I=

∫∞

−∞

dx e−ax

2

.

Integrate over bothxandyso that

I^2 =

∫∞

−∞

dx e−ax

2

∫∞

−∞

dy e−ay

2

=

∫∞

−∞

∫∞

−∞

dxdy e−a(x

(^2) +y (^2) )
.
Transform to polar coordinates.
I^2 = 2π

∫∞

0

rdr e−ar

2

=π

∫∞

0

d(r^2 )e−ar

2

=π

[

−

1

a

e−ar

2

]∞

0

=

π

a

Now just take the square root to get the answer above.

∫∞

−∞

dx e−ax

2

=

√

π

a

.

Other forms can be obtained by differentiating with respect toa.

∂

∂a

∫∞

−∞

dx e−ax

2

=

∂

∂a

√

π

a

∫∞

−∞

dx x^2 e−ax

2

=

1

2 a

√

π

a