Future learning for a math graduate in applied mathematics references

Question

As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications for given statements.

So for my future work i would like to learn some of the applications (mostly in electronics, programming, physics...) and get some references where i can continue to learn so that - given a problem i can correlate that to something i have learned before.

I am most interested in applications for these fields:

1. Numerical analysis

   • Interpolation (Hermite, multidimensional, Newton, Spline ...)
   • Approximation ( Least squares, uniform approximation,..)
   • Numerical methods for solving differential equations
   • Numerical methods for finding eigenvalues and eigenvectors

   • Numerical Methods for solving non-linear equations

     ...

2. Mathematical analysis

   • Integration by curve, surface
   • Integrals with parameters
   • Fourier series,Fourier transformation,Fourier integral
   • Uniform convergence (for sequence of functions, series, integrals)

   • Weierstrass function, Riemann zeta function

     ...

3. Measure theory

   • Lebesgue measure, Lebesgue integral
   • Radon - Nikodym theorem , derivate
   • Monotone convergence theorem, dominant convergence theorem

   • Lp spaces, norms

     ...

4. Complex analysis

   • Cauchy integral theorem
   • Picard's theorem
   • Laurent series ...

   • Holomorphic functions

     ...

References, books, websites - anything will do, just as long as there are multiple examples (also the more on the side of programming (algorithms, problem solving) and electronics the better)

Answer 1

Let's check out the OP's wish list:

1. Numerical analysis

   • Interpolation (Hermite, multidimensional, Newton, Spline ...)
   The webpage Programming in Delphi contains some material (theory & software) about interpolation, with 2nd order splines only.
   • Numerical methods for solving differential equations
   My own experience in this field has been summarized as : Highlights .
   (Link, because I don't want this answer look like personal advertizing)
   Research concerns the linear partial differential equations for convection / diffusion.
   And a Least Squares (Finite Element) Method is employed with ideal flow.
   Some of my favorite readings:
   • Numerical Heat Transfer and Fluid Flow by Suhas V. Patankar
   • Books by O.C. Zienkiewicz (Finite Element Method)
   Warning. The mathematics in such practical applications (engineering) books may be very specific, i.e not so "general" as is common in "theoretical" treatments. It may contain a lot of hand waving argumentation as well. Looking through all this is part of the challenge. What's more, the two main competitors in Numerical Analysis, the Finite volume method and Finite element method, have been invented by engineers in the first place, not by mathematicians. The latter have jumped in (especially with FEM) only after the beef has been roasted already.

2. Mathematical analysis

3. Measure theory

Somebody somehow has to show me the indispensability of Measure theory for practical applications. Of course, in some of the comments, Probability theory is mentioned as the counter example par excellence. Then we have the following MSE question as a reference: Between bayesian and measure theoretic approaches .

4. Complex analysis

An exellent book is: Complex made Simple , written by one of the top mathematicians in this forum: David C. Ullrich. (The book is somewhat less simple than it sounds, though)-: EDIT. And indeed, there is no specific Application part in it. So here comes a list of Q & A in this forum that is maybe more relevant for the OP:
• Find an analytic function that maps the angle $-\pi/4<\operatorname{arg}(z)<\pi/2$ onto the upper half plane so that $w(1-i)=2,w(i)=-1$, and $w(0)=0$
• a problem in application of conformal mappings
• Find the velocity field of ideal fluid that has a sink strength of $2\pi k$ using complex analysis.
• How to differentiate Complex Fluid Potential
• Applications of conformal mapping
• How to differentiate Complex Fluid Potential
• CMS : Complex Made Simpler (?)

Answer 2

Probability theory is to me strictly pure mathematics, but with strong applicable properties. If you like measure theory, along with functional/modern analysis then probability theory is the area for you; The axioms of probability theory (Kolmogorov) are strictly measure theoretic. Then how one builds up expected value and so on is also integration theoretical. One integrates with respect to a probability measure, thus making the Lebesgue integral or equivalent necessary. I strongly encourage you to read the table of content in Kallenberg's Foundations of modern probability theory, to get a glimpse of what probability theory might entail, from the analytical stand point. It is sadly a bit advanced though, but I do not expect someone to read it all.

Probability theory is to some degree either very analytical or very combinatorial/graph theoretical, i.e. either you get hooked on stochastic analysis and SDE:s (along with stochastic integration and such), or you get hooked on percolation theory and Markov analysis, the latter having intimate connections with statistical mechanics (Ising's model for instance).

And as a last point, probability theory is computationally challenging, in so far as we need strong programmers as well. For instance, Monte Carlo methods are a very popular numerical method.

I hope you do not mind me diverging from your wish list, but I believe probability theory is worth a notice.