PYL113: MATHEMATICAL PHYSICS

TUTORIAL SHEET 3

1. Prove the following property in the context of Fourier transformation:

FTZ t

f(s)ds

=

1

i!f(!) + 2c(!),

where c is the constant of integration.

2. By applying Fourier Inversion theorem prove that

2exp( | t |) =

Z 10

d!cos!t

1 + !2.

3. Find the Fourier transform of

f(t) =

(1, | t |< 1,0, otherwise.

(a) Determine the convolution of f with itself and without further

integration, deduce its transform.

(b) Deduce that Z 11

sin2 !

!2d! = ,Z 1

1

sin4 !

!4d! =

2

3.

4. In class, we saw that the Fourier transform of a Gaussian is also a

Gaussian. Another frequently encountered function is a Lorentzian:

L(x) =1

x2 + 2.

Obtain the Fourier transform of this function.

5. A semi-infinite rectangular metal plate occupies the region 0 x 1and 0 y b in the xy-plane. The two long sides and the far end of

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the plate is fixed at 0oC and the x = 0 edge is at a temperature uo.

The steady state distribution is given by

u(x, t) =1Xn=1

Bn exp(nx/b) sin(nx/b).

Evaluate the constants Bn.

6. (a) Find the Fourier transform of

f(, p, t) =

(et sin pt, t > 0,

0, t < 0,

where (> 0) and p are the constants.

(b) The current I(t) flowing through a certain system is related to the

applied voltage V (t) by the equation

I(t) =

Z 11

K(t u)V (u)du,

where K(t) = a1f(1, p1, ) + a2f(2, p2, ). The function f(, p, t)

is as given in (a) and all the ai, i (> 0) and pi are fixed parameters.

By considering the Fourier transform of I(t), find the relationship

that must hold between a1 and a2 if the total net charge Q passed

through the system (over a very long time) is to be zero for and

arbitrary applied voltage.

7. A linear amplifier produces an output that is the convolution of its

input and its response function. The Fourier transform of the response

function for a particular amplifier is

eK(!) = i!p2 ( + i!)2

.

Determine the time variation of its output g(t) when its input is the

Heaviside step function.

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Rachit Madanfourier seriesuse orthoganility

8. For some ion-atom scattering processes, the potential is given by:

V =| ~r1 ~r2 |1 exp ( | ~r1 ~r2 |) .

Show, using the worked example in subsection 13.1.10, that the prob-

ability that the ion will scatter from say, ~p1 to ~p2 is proportional to

(2 + k2)2 where k =| ~k | and

~k =1

2~ [(~p2 ~p1) (~p1 ~p2)] .

9. Calculate directly the auto-correlation function a(z) for the product of

the exponential decay distribution and Heaviside step function

f(t) =1

etH(t).

Use Fourier transform and energy spectrum of f(t) to deduce thatZ 11

ei!z

2 + !2d! =

e|z|.

10. The one-dimensional neutron diusion equation with a (plane) source

is

Dd2(x)

dx2+K2D(x) = Q(x),

where (x) is the neutron flux, Q(x) is the source at x = 0, and D

and K2 are constant. Obtain the solution of the dierential equation

using the Fourier transform technique.

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