Math Methods for Polymer Science
Lecture 2: Fourier Transforms, Delta Functions and
Gaussian Integrals
In the first lecture, we reviewed the Taylor and Fourier series. These
where both essentially ways of decomposing a given function into a different, more convenient, or more meaningful form. In this lecture, we review
the generalization of the Fourier series to the Fourier transformation. In the
context, it is also natural to review 2 special functions, Dirac delta functions
and Gaussian functions, as these functions commonly arise in problems of
Fourier analysis and are otherwise essential in polymer physics. For additional reading on Fourier transforms, delta functions and Gaussian integrals
see Chapters 15, 1 and 8 of Arken and Weber’s text, Mathematical Methods
for Physicists.
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Fourier Transforms
Conceptually, Fourier transforms are a straightforward generalizations of
Fourier series which represent a function on finite domain
of size L by
an
2πn
2πn
infinite sum over a discrete sets of functions, sin L x and cos L x . In
the Fourier transform, that size of the domain is taken to ∞ so that the
domain becomes the all positive and negative values of x (see Fig. 1). The
key differences between Taylor series, Fourier series and Fourier transforms
are summarized as follows:
Taylor Series - series representation in polynomials, “local” representation
Fourier Series - series representation in sines and cosines, “global” representation (over finite or periodic domain)
Fourier transform - integral representationin sines and cosines over infinite domain (L → ∞)
How do we generalize a Fourier series to an infinite domain? For the
Fourier transform it’s more convenient to use complex representation of sine
and cosine:
eix = cos x + i sin x
(1)
1
Figure 1: Schematic of Fourier transform of function f (x), where f (x) =
X
∞
∞
a0 X
2πn
2πn
+
x +
x .
an cos
bn sin
2
L
L
n=1
n=1
Using this we can rewrite the Fourier series:
∞
a0 X
an cos
f (x) =
+
2
n=1
=
∞
X
cn exp
n=−∞
X
∞
2πn
2πn
x +
bn sin
x
L
L
n=1
i2πnx
L
where
(
cn =
(2)
an − ibn
2
an + ibn
2
a0
c0 = .
2
n>0
(3)
n<0
(4)
Notice also that complex functions ei2πnx/L , are also orthogonal.
Z
L
2
−L
2
dx exp
L/2
i2πn
L
2π(m + n)i
i2πm
x exp
x =
exp
x L
L
2π(m + n)i
L
−L/2
L
π(m + n)
sin
π(m + n)
L
0
for m + n 6= 0
=
L
for m + n = 0
=
(5)
Using this, we can extract Fourier coefficients by
1
cn =
L
Z
L
2
−L
2
2πn
dx exp −
x f (x)
L
Note that the complex notation takes care of factors of 2, etc.
2
(6)
We want to define Fourier transform as the L → ∞ limit of a Fourier
series. This limit is unusual because we take L → ∞ while
2πn
≡k
L→∞ L
lim
(finite).
(7)
Here, k is referred to as the wavenumber of the Fourier mode, eikx . The
Fourier transform of a function, f (x), is defined as:
Z ∞
˜
f (k) = lim (cn L) =
dx e−ikx f (x).
(8)
L→∞
−∞
Often the Fourier transform is written as
F [f (x)] = f˜(k)
(9)
where F means Fourier transform. Notice that the Fourier transform takes
f (x), function of “real space” variables, x, and outputs f˜(k), a function of
“Fourier space” variables, k.
The Fourier transform can be inverted using the definition of the Fourier
series
∞
∞
X
X
1
cn e−ik(n)x =
f (x) =
(cn L)eik(n)x .
(10)
L
n=−∞
n=−∞
Now as L → ∞ k takes on a continuum of values
lim ∆k = k(n + 1) − k(n) =
L→∞
2π
2π
[(n + 1) − n] =
→0
L
L
That is, δk becomes infinitely narrow in the L → ∞ limit. This means we
can write the sum as
Z ∞
∞
∞
X
X
1
1
1
lim
=
lim
∆k =
dk
(11)
L→∞
L
2π ∆k→0 n=−∞
2π −∞
n=−∞
The last step is the is Riemann’s definition of an integral (see Fig. 2). And
thus we get
Z ∞
dk ˜
f (x) =
f (k)eikx
(12)
−∞ 2π
Eq. (12) is the inverse Fourier transform or
h
i
F −1 f˜(k) = f (x)
(13)
where F −1 means inverse Fourier transform, f˜(x) is function of “Fourier
space”, and f (k) is function of “real space”.
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Figure 2: A figure showing that Fourier series becomes an integral of continuous function f˜(k) in limit that ∆k → 0 (area).
1.1
Delta Function
Related to the Fourier transform is a special function called the Dirac delta
function, δ(x). It’s essential properties can be deduced by the Fourier transform and inverse Fourier transform. Here, we simply insert the definition of
the Fourier transform, eq. (8), into equation for the inverse transform, eq.
(12),
Z ∞
Z
Z ∞
dk ˜
dk ikx ∞ 0 ikx0
ikx
f (x) =
dx e f (x0 )
f (k)e =
e
−∞ 2π
−∞
−∞ 2π
Z ∞
Z ∞
dk ik(x−x0 )
=
dx0
e
f (x0 )
2π
−∞
Z−∞
∞
=
dx0 δ(x − x0 )f (x0 ) = f (x)
(14)
−∞
This last line defines the properties of delta function, which is defined implicitly by the integral in the parentheses on the second line. When you
integrate the product of the Dirac delta function with another function, it
returns the value of that function at the point where the argument of δ(x)
vanishes. Geometrically, you can think of it as an infinitely tall and narrowly
peeked function, with area 1 under the curve (see Fig. ??).
Using this definition of δ(x) we can derive the Fourier transform of oscillatory functions.
4
Figure 3: Sketch of a Dirac delta function.
Figure 4: Plot of f (x), which is only non-zero near x = 0.
Example 1: Compute Fourier transform of A cos(qx).
Z ∞
˜
f (k) =
dx A cos(qx)eikx
−∞
Z
A ∞
=
dx eiqx + e−iqx eikx
2 −∞
Z
h
i
A ∞
=
dx ei(q+k)x + e−i(q−k)x
2 −∞
= πA [δ(q + k) + δ(q − k)]
which is only non-zero for k = ±q.
The Fourier transform is particularly useful for studying the properties of
a function which is non-zero only over a finite region of space. For example,
density or probability distribution for polymer chain.
For this case, we can use Taylor series expansion to cast light on what
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Figure 5: Plot of f (x).
the Fourier transform tells us.
Z ∞
˜
f (k) =
dx e−ikx f (x)
−∞
Z ∞
i 3 3
1 4 4
1 2 2
=
dx 1 − ikx − k x + k x + k x + . . . f (x)
2!
3!
4!
−∞
Clearly,
f˜(k = 0) =
(15)
∞
Z
dx f (x)
(16)
−∞
which is the total area under curve, named N. But we see from Taylor series
that various powers of k represent certain averages. That is,
1 2 2
i 3 3
1 4 4
˜
f (k) = N 1 − ikhxi − k hx i + k hx i + k hx i + . . .
(17)
2!
3!
4!
where
Z
∞
dx xn f (x)
−∞
hxn i = Z
∞
(18)
dx f (x)
−∞
is an average of xn weighted by f (x).
That is what I mean when we say that f˜(k) is something of a global
representation. f˜(k) seems to encodes properties of the function over its
entire range not just locally.
Let’s try an example. Compute Fourier transform of
A
|x| ≤ a
f (x) =
0
|x| > a
6
Figure 6: Plot of Gaussian function.
f˜(k) =
Z
∞
−ikx
dx f (x)e
Z
a
=A
−∞
dx e−ikx
−a
i 2A sin(ka)
A h −ika
e
− eika =
=
ik
k
(19)
Now that we have full expression, let’s examine small k behavior.
ka − 3!1 (ka)3 + . . .
sin(ka)
lim f˜(k) = 2A
= 2A
k→0
k
k
1
= 2Aa 1 − (ka)2 + . . .
3!
(20)
This is just the form we derived above. Note that 2Aa = N (area), and you
can check that
Z a
1 3 a
2
x
dx x
3 −a
a2
−a
2
hx i = Z a
=
=
(21)
2a
3
dx
−a
1.2
Gaussian Integral
Let’s use the Fourier transform to study an important function, the Gaussian
bump
x2
f (x) = Ae− 2a2
7
(22)
This function is very important in random systems, especially in polymer
physics.
Aside: Integrating a Gaussian function (A trick!).
Z ∞
x2
dx exp − 2
(23)
I=
2a
−∞
2
I =
2
2 Z ∞
Z ∞
x2
x + y2
dx exp − 2
=
dx
dy exp −
2a
2a2
−∞
−∞
−∞
Z
∞
(24)
This double integral is carried outpover whole x − y plane. Let’s do same
integral in polar coordinates: r = x2 + y 2 , x = r cos φ and y = r sin φ.
Z ∞
Z 2π
r2
2
dr r exp − 2
dφ
I =
2a
0
0
Z ∞
d
r2
2
dr
= 2π
−a exp − 2
dr
2a
0
∞
r2
= 2π −a2 exp − 2
(25)
= 2πa2
2a
0
√
thus, I = 2πa.
Let’s go back to Fourier transform of a Gaussian bump.
Z ∞
2
2
˜
f (k) = A
dx e−x /2a e−ikx
(26)
−∞
This can be done by ”completing the square” of argument in exponential
1
k 2 a2
x2
2 2
+
ikx
=
(x
+
ika
)
+
2a2
2a2
2
(27)
then
∞
2 2
(x + ika2 )2
k a
dx exp −
exp −
2
2a
2
−∞
2 2Z ∞
k a
u2
= A exp −
du exp − 2
2
2a
−∞
2 2
√
k a
= A 2πa exp −
2
f˜(k) = A
Z
(28)
Note that this was done by changing variables u = x + ika2 and du = dx.
Once again we can learn something of f (x) by examining small k properties of f˜(k).
√
k2
k4
f˜(k) = Aa 2π 1 − a2 + (3a2 ) + . . .
(29)
2!
4!
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Figure 7: Schematic of an electron density distribution.
There is no odd terms since hxn i = 0 for n odd since f (x) = f (−x). By
taking Fourier transform of Gaussian function, we have automatically calculated all moments, or averages, of the distribution. The most important
is the second moment
Z ∞
x2
dx exp − 2 x2
2a
= a2
hx2 i = −∞
(30)
Z ∞
x2
dx exp − 2
2a
−∞
We’ll use this extensively.
Finally, we conclude our discussion of Fourier transforms with a discussion of convolution.
Z ∞
f ∗g =
dy f (y)g(x − y)
(31)
−∞
where f ∗ g stands for convolution of f (x) and g(x), which are function of x
only.
Convolution theorem says
F {f ∗ g} = f˜(k)g̃(k) = F {f }F {g}
(32)
Fourier transform os convolution is product of Fourier transform of f (x) and
g(x).
This is fairly straight forward to prove. Insert
Z ∞
dk ik(x−y)
g(x − y) =
e
g̃(k)
(33)
−∞ 2π
above and take Fourier transform.
What is this good for? Say you have a density (probability) distribution
created by a lot of copies of the same thing (identical proteins floating in
solution). For example, a given macromolecule created a Gaussian electron
distribution around it’s center of mass.
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The total density is computed by summing individual distributions around
position of each molecule.
ρcm (x) = δ(x − x0 ) + δ(x − x1 )
where ρcm (x) is density of molecule center of mass, and
Z ∞
dy ρcm (y)ρcm (x − y)
ρtot (x) =
(34)
(35)
−∞
Fourier transform (needed form scattering) is simply product of individual Fourier transforms.
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