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PHY325H5F: Mathematical and Computational Physics

Problem Set 3


1. (20 marks) Fourier Transforms and Parseval’s theorem

(a) Use the fact that (known as Fourier’s inversion theorem) to prove that the following is true. (/10)

Hint: First find the Fourier transform of

Recall: and 

(b) Demonstrate the application of Parseval’s theorem for the exponential function . (/10)

Hint: The substitution will help with the integration.


2. (20 marks) Consider the differential equation of a damped harmonic oscillator sub-jected to a force :

where , and

(a) Find an expression for  (/10)

(b) We can relate the output of a system to some external input by the expression:

where is the linear response function. Using the convolution theorem, find an expression for . Express your final answer as a complex number of the form , where and are real variables (they don’t have any complex numbers). In other words, you should not have any complex numbers in the denominator. (/10)

The expression you obtained describes the response of the harmonic oscil-lator to a force which results in the observed motion in the ”Fourier domain”. The imaginary part of  is related to energy dissipation. If you could calculate the inverse Fourier transform, you would get the response of the system as a function of time - but that’s a story for another time...


3. (40 marks) Using the convolution theorem in conjunction with Parseval’s theorem show the following (note: do not directly perform the integrations):

(a) Find the Fourier transform of the unit rectangular distribution(/5):

(b) With the results of (a) use Parseval’s theorem to show the following(/10):

(c) First determine the convolution of with itself, i.e. , then use Parseval’s theorem to show the following(/25):