PH 223: Complex Analysis and Integral Transforms (2023) Autumn
Prof Dibyendu Das
Prerequisite
None
Course Content
Part A (60% of the course): Complex analysis 12) Complex numbers z. Complex plane. Triangle inequalities. 13) Continuity, Differentiability, Continuity and existence of partial derivatives, Cauchy-Riemann conditions, pure complex function of z. 14) Analyticity. Single value. Cauchy’s theorem. Complex Taylor Series. Convergence and domains of analyticity. Order of zeros. 15) Cauchy integral formula, and derivation of Laurent Series. Calculation of Residues. 16) Meromorphic functions and order of poles, Branch singularities, Essential singularities. 17) Residue theorem. Cauchy’s argument principle. 18) Various types of contour integrals — semicircles, rectangular, conical. Case of poles on real line. 19) Branch Points and branch cuts. Integrals involving branch singular integrands. 20) Shapes of complex functions and Saddles. Darboux’s inequality. Proofs of impossibility of local maxima and minima. Impossibly of entire & bounded function. 21) Asymptotic analysis: Laplace’s method, stationary phase, and method of steepest descent. 22) Conformal Mapping and properties. Linear and Inversion map and their geometric effects on lines and circles. Application to 2d electrostatics. Logarithmic map. Homographic transformations and cross-ratio preservation. Part B (40% of the course): Function spaces, Integral transforms and some Differential equations 7. Infinite dimensional vector spaces or function spaces. Inner product, and weight function. The problem of completeness. Riemann to Lebesgue integrals, and Lebesgue space. Reisz-Fisher theorem. Bessel inequality, and Perseval’s equality. Hilbert space. 8. Weierstrass’s theorem and polynomial basis. Orthonormalization of polynomials using Schmidt method. Generalized Rogrigues’s formula and 3 classes of classical polynomials. 9. Good, fairly good, and generalized distributions (namely, Dirac delta). Continuous index basis and use of Dirac delta. Identity operator and completeness relation of polynomials. 10. Fourier series as a basis expansion. Fourier cosine and sine series. Fourier transforms. Plancherel-Parseval relations. Meaning of Fourier transforms of generalized functions like Dirac delta where Parseval relation fails. 11. Examples of Fourier transform calculations — reminding complex analysis. Transforms of derivatives and derivatives of transforms. Solving linear ODEs, and PDEs (like diffusion equation) using Fourier transform. Convolution theorem. 12. Laplace transforms. Examples. Derivatives of transforms and transforms of derivatives. Shifting properties. Solution of linear ODEs and PDEs. Convolution theorem. Inverse Laplace transforms, Bromwich integrals and contour integration.
Books
Mathematics for PhysicistsP. Dennery and A. Krzywicki Dover BooksMathematical Methods for PhysicistsG.B.Arfken and H. J. WeberElsevier Press
Review by Arnav Jain
Lectures
Lectures are fast, but taught well. You will need to grapple with the concepts at times, as it may not be obvious to you in lectures, but a revision of notes will help. The tutorials provided are lengthy, but very satisfactory on completing. They are also a great template; most of the questions in exam are based on the questions in tutorial, with a slight twist at times. The professor also gives and explains the solutions in class, so make sure you write them down as they are very handy! The mathematics explained in this is pretty useful if one goes forward in the theoretical, and is not as rigorous as a MA course.
Assignments, Exams and Grading
Two quizzes, a midsemester and an endsemester. The endsemester has a higher weightage than most theory courses. The exams contain all variety of questions, from easy to tricky. However, they do tend to be lengthy. Grading was average.
Tips
Consistent effort is needed for this one; it is a fast course.
Review by Aditya Saran
Lectures
The course is really interesting and the content of this course will help you everywhere in Physics. The way of teaching of the professor is also excellent. The best way to get good grades in this course is to solve the assignment very thoroughly.
Assignments, Exams and Grading
1st Quiz (10%) - Moderate , Midsems (25%)- Difficult, 2nd Quiz (15%) - Moderate, Endsems(50%)- Difficult
Tips
Many a times assignment questions could be very lengthy but make sure you do all questions because the instructor makes every question so as to teach you a new concept.
Review by Yashowardhan
Lectures
This course is extremely well lectured by Dibyendu sir, he makes classes interesting by providing useful context wherever he can and his explanations flow in a musical manner. His notes are very detailed and problem sets very rich with nuance and hidden learning, so going through each line of his material is paramount for absorbing everything from the course. This course will push your calculative skills to the max, and whilst having a theoretical understanding and overview of the content taught is useful, at the end of the day lot of practice is required to do the middling difficulty questions in his exams, which are handmade by the prof so don't expect the tutorials to be enough, as he has a habit of twisting several concepts into one question.
Assignments, Exams and Grading
1st Quiz (10%) - Moderate , Midsems (25%)- Difficult, 2nd Quiz (15%) - Moderate, Endsems(50%)- Difficult
Tips
Do not miss lectures, he puts alot of thought into explaining during the class and its very difficult to reconstruct all of it from just the notes, although once you've understood the concepts his notes are very good. If you've ended up missing two lectures in a row or more, get back upto pace with class fast otherwise cramming even several days before the exam will not recover your score in his exams. He asks questions from even smaller topics and tends to leave no stone unturned in the exams, so build a good understanding of all topics if you're not perfect in all.
Review by Disha Zaveri
Lectures
A really interesting course with concepts that will help build a base in understanding later topics as well. Professor Dibyendu Das explains concepts really well and goes into enough mathematics to give a good intuition about the topics without feeling overwhelming. However, he goes a little fast at times so it's important to pay close attention in his classes to not miss something. The assignments are very carefully designed, with each question highlighting different concepts. Solving them gives greater clarity and appreciation for the subject, but they can be difficult to approach at times.
Assignments, Exams and Grading
1st Quiz (10%) - Moderate , Midsems (25%)- Difficult, 2nd Quiz (15%) - Moderate, Endsems(50%)- Difficult 1 page cheat sheets was allowed No attendance criteria
Tips
Attend classes, they help give at least a basic understanding of the course and make it easier to understand the notes. Make sure you've solved the assignments thoroughly, and preferably on your own. The exams, especially the quizzes, will become much easier this way as most questions are similar to some in the assignment itself. It's important to have practice in mathematical techniques (integration, common tricks, etc) as well, because questions are often lengthy and rely on calculations once the initial concept is understood. Do not try and cram one day before the exam (a mistake I made too often). Consistency is key to doing well in this course.
Review by Sachin raj
Lectures
It was a nice course. This course is very useful especially for Physics students as almost in every branch of Physics complex integrals, integral transforms .... come
Assignments, Exams and Grading
All quizzes, midsem and endsem exams were good. There were not any marks for attendance. All the assignments had very good questions. There were at least 15-25 questions in each assignments. Tests were closed book but cheats sheets were allowed. One cheat sheet was allowed for midsem and quizzes and two cheat sheets were allowed for endsem exam.
Tips
Mathematical methods for Physicists by George B.Arfken and Hans J.Weber
Review by Anonymous
Lectures
The course content was taught in a fairly intuitive manner and those who were regular with their attendance and solving the assignments, benefitted the most. Mathematical rigor (not to be confused with complexity) was sacrificed sometimes in order to be able to cover a broader range of topics and carry out increasingly complicated calculations. This can lead to practise having a more important role to play than understanding in some cases. Professor took care not to make the assessments and assignments too easy. A decent amount of practise and good skill in algebraic manipulations was required to be able to do the questions and it was a must to have the notes in one's head completely.
Assignments, Exams and Grading
1st quiz (easy-moderate) – 10%, Midsem (most difficult) – 25%, 2nd quiz (moderate) – 15%, Endsem – 50% (moderate-difficult).
Tips
Schaum's Outlines – Complex Variables (for up to the Contour Integrals part). Mathematical methods for Physicists by George B. Arfken and Hans J. Weber. Doing at least one contour integral problem per day (not as easy as it sounds. Requires a remarkable level of consistency) will shed a lot of the initial burden of the course. The 1st quiz was arguably easier than the 2nd quiz, but people's scores on an average rose with each exam. This implies, as time went by, people started taking the course more seriously. That period needs to be cut out and the course needs to be taken seriously from the get-go. The techniques will prove to be useful and important to be proficient in, for later courses as well. Even those who don't have a habit of writing should write and practise the assignment problems of this course, as its main issue is not being conceptually challenging (not to imply that this challenge doesn't exist).