Mathematical Physics 01 - Carl Bender

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Strong rec Daniel R · 2013-06 · Mathematics Stack Exchange · on “Approximation of differential equations”

As a reference, check out Advanced Mathematical Methods for Scientists and Engineers [LINK] by Carl Bender and Steven Orszag. It handles approximate solutions of both linear and nonlinear differential equations. Among other things, Perturbation Theory [LINK] is discussed thouroughly. It involves turning a difficult problem (a hard to crack ODE for example) into an infinite amount of easy problems leading to an asymptotic solution. Also, check out the excellent lecture series on the subject by the above author on [this video] .

Strong rec TheQuantumMan · 2015-11 · Physics Stack Exchange · on “Is there an online course in Mathematical Methods for Physics, Covering Matrices and Vector Analysis?”

I only know of these: [this video] (PERIMETER INSTITUTE RECORDED SEMINAR ARCHIVE) [LINK] (Indian University of Technology Madras) The first two I recommend because they are simply the same courses as the one that you are attending. The following are pure mathematics but I know that their quality of teaching is very good and also cover most of the material that you described. [LINK] (UC Berkeley) [LINK] (MIT by Gilbert Strang who is considered by many to many a very good teacher) [LINK] (MIT- this is my personal favorite!)

Recommends Daniel R · 2013-06 · Mathematics Stack Exchange · on “Approximation of differential equations”

Have you had a look at Perturbation theory [LINK] ? Check out Carl Benders lectures in Mathematical Physics [this video]

Recommends Dilaton · 2014-07 · Physics Stack Exchange · on “Ambiguity in Asymptotic Perturbative Series and Instantons”

Some explanations are given here [LINK] and Prof. Carl Bender's Mathematical Physics course [this video] , which exactly deals with asymptotic series, might be helpful.

Recommends Apoorv Potnis · 2022-06 · Physics Stack Exchange · on “Is there any book that treat time-dependent perturbation theory with rigorous mathematics?”

I don't know if this book contains what you are looking for but might help you: _Advanced Mathematical Methods for Scientists and Engineers_ vol. 1 by Bender and Orszag. [LINK] Lecture course by Prof. Bender [this video]

Cites Leo · 2016-05 · Mathematics Stack Exchange · on “Why is $\varepsilon x^5 \sim -x$?”

I'm trying to understand what's going on in this lecture on perturbation [this video] (the link brings you to 1h 08m 12s). The original problem is to find the real root of $$x^5+x=1.$$ We have inserted $\varepsilon$ as follows $$\varepsilon x^5+x=1$$ and are discussing which of the three terms ($\varepsilon x^5$ or $x$ or $1$) is negligible as $\varepsilon \to 0$ so we could throw it way. So we consider all three cases and it turns out that: Term $1$ is negligible and we learn something if we remove it, Term $x$ is not negligible, Term $\varepsilon x^5$ is negligible but we learn nothing if we remove it. My question is about term $1$ being negligible. On the blackboard, we have $$\varepsilon x^5

Cites Brightsun · 2017-04 · Mathematics Stack Exchange · on “Perturbation series for $x^5+\varepsilon x-1=0$”

I want to find a closed form for the perturbation coefficients $a_n$ defined by the perturbative solution $$ x(\varepsilon)=1+\sum_{n=1}^\infty a_n \varepsilon^n $$ to the quintic equation $$ x^5+\varepsilon x-1=0. $$ By computing the determinant, as was suggested for this related question [LINK] regarding the cubic case, we can argue that the radius of convergence of the above series must be $$ \rho=\frac{5}{4^{4/5}}=1.64938\dots $$ Furthermore, the Lagrange-Bürmann theorem allows us to formally write down a closed form for the coefficients, namely $$ a_n=\frac{(-1)^n}{n!}\frac{d^{n-1}}{dx^{n-1}}\left(\frac{x}{1+x+x^2+x^3+x^4}\right)^n_{x=1} $$ but this doesn't look very illuminating. (For

Cites spraff · 2020-07 · Mathematics Stack Exchange · on “What does it mean when polynomials have closed, exact complex solutions, but not exact real solutions?”

I was watching this introduction to peturbation theory [this video] . His first example is solving $$x^5 + x = 1$$ for which he claims there is no exact real solution. I asked WolframAlpha [LINK] what it thought. It gives an inexact decimal solution $x \approx -0.75488...$ and some exact complex solutions $$x = -\sqrt[3]{-1}$$ $$x = (-1)^\frac{2}{3}$$ [LINK] [LINK] Is there some deep reason as to why the complex roots would have exact forms but not the real root? Could we have an $n$ -degree polynomial with $a$ exact solutions and $b$ inexact solutions, for arbitrary $a+b=n$ ? Can the exact and inexact solutions be distributed arbitrarily between the real line and the rest of the complex plane? C

Cites Bijou Smith · 2025-01 · Physics Stack Exchange · on “Reason for the discreteness arising in quantum mechanics?”

A few answers offered are either (a) not valid and/or (b) are not answering the question. For issue (a): It is not the mathematics that determines the physics, the mathematics only models the physics. For issue (b): The question is best answered by the reply that noted bound states are where energy levels are discrete, or in systems with semi-bounding potentials, like a wall. The Postulate of QM responsible is the superposition postulate. This is why wave forms arise as solutions to the spinor equation, which is essentially describing stochastic dynamics. (The result is the probability density.) On this interpretation it is not that energy levels are discrete per se , but that the transition