Tutorato Metodi Matematici della Fisica

Here you can find the various notes of the lectures with the exercises presented during the lessons plus some other exercises that I didn’t have time to do in class. Most of the notes come with some basic theory behind the exercises themselves.

The lectures are held both online and in presence in the following days

I strongly recommend to follow the lectures in class so that you can ask me any question directly. For any other questions related to exercises, done and not done during the lecture, you can contact Me or Marco Marino at the following e-mail addresses

• Davide Morgante: davide.morgante@mi.infn.it
• Marco Marino: marco.marino1@unimi.it

The number of my office is DC/T/37 and is located at the ground floor, at the end of the central corridor to the left, near the bathrooms (the office is two doors from prof. Molinari’s office). For the location of Marco’s office, write him an e-mail!

Important announcement: last lesson was on 16/06/2022. If you need help before the exam just come to our offices. I recommend to write an e-mail some days before to know if we are available. In any case, you can knock on our doors, maybe you’ll find us in the office!

Lecture 1: Complex differentiability, Holomorphic functions, Harmonic functions, Möebius Transformaion (04/04/22)

You can find the pdf of the lesson from this link.

Lecture 2: Cauchy integral formula, Taylor and Laurent series, Radius of convergence, Cayley-Hamilton theorem for the power of a matrix (07/04/22)

You can find the pdf of the lesson from this link. Here you can find some notes on Cayley-Hamilton theorem.

Please Note: There is an error in the solution of the Cayley-Hamilton exercise. I want to thank the students for making me notice it. You can find the right solution from this link.

Lecture 3: More Taylor and Laurent series (11/04/22)

The second method to do the series expansion of $f(z) = (z^2-5z+4)^{-1}$ using Cauchy product formula, can be found in the lesson 2 notes.

The notes for the big exercise done during class can be found from this link.

Lecture 4: Residue Theorem, Cauchy Principal Value, Order $k$ poles and their residues, Integrals with branch cuts (28/04/22)

You can find the pdf of the lesson from this link. Check out the last exercise that we didn’t have time to do. You are asked to integrate the function \(f(z) = \frac{e^\frac{z}{2}+|z|^2}{(z-i\pi)^2}\) on the circular contour centred at $z=i\pi$ with radius $R=\pi$. Not only this function has a second order pole, but you cannot directly use the residue theorem on the whole function since is not holomorphic. This exercise is taken from an exam, so be careful! Moreover, in this exercise I give you two ways of evaluating a second order pole: the first with the generalised formula that we have seen for the first order pole and the second (the best one if you ask me) is through the Laurent expansion of the function. If you have any questions, just write me an e-mail.

Lecture 5: More residue integrals (02/05/22 - Marco)

You can find the pdf of the lesson from this link. There are more exercises on the pdf which Marco will probably do on Monday 09/05.

Lecture 6: Guess what: more residue integrals (05/05/22)

You can find the pdf of the lesson from this link.

Please Note: When doing the integral on the small circle path around $z=0$ and taking the limit $r\rightarrow 0$ the limits of integration when we parametrise the curve are $[0,\pi]$ and not $[0,2\pi]$ as I did in class. Sorry for the inconvenience!

Lecture 7: Probably more integrals (09/05/22 - Marco)

You can find the pdf of the lesson from this link.

Lecture 8: Residue Integrals & Möbius transformations (13/05/22 - Marco)

You can find the pdf of the lesson from this link.

Lecture 9: Finite dimensional Hilbert Spaces and Operators acting on such Hilbert spaces (16/05/22)

You can find the pdf of the lesson from this link

Lecture 10: Operators on finite dimensional Hilbert spaces, Group theory and Lie Groups (19/05/22)

You can find the pdf of the lesson from this link.

If you want to see how to solve the exercise on the generator of the representation of $SO(1,1)$ without knowing anything about Lie groups and algebras, you can find the solution from Molinari’s notes here.

Lecture 11: Infinite dimensional, separable, Hilbert spaces and Operators on them (23/05/2022)

You can find the pdf of the lesson from this link.

Lecture 12: Infinite dimensional Hilbert spaces, $L^2(X)$ spaces and separability (26/05/2022)

We started the lesson with the last exercise of the last pdf. The other exercises can be found from this link.

Lecture 13 & 14: Fourier Series & Fourier Transform (30/05/2022 - 06/06/2022 - Marco)

You can find the pdf of the lessons from this link.

Lecture 15: Tempered Distributions (09/06/2022)

You can find the pdf of the lesson from this link.

Lecture 16: Operators & Hilbert spaces (13/06/2022 - Marco)

You can find the pdf of the lesson from this link.

Lecture 17: Fourier Transforms and Fourier Transform Equations (16/06/2022) - Last lecture!

You can find the pdf of the lesson from this link. Note that this pdf was written by me many years ago, while I was following this course so there are going to be many errors, be careful!