# Are there more modern or computation oriented applications of complex analysis in science and engineering?

No doubt that complex analysis is a tremendously useful with plenty of applications in engineering and physics.

Common raw applications of complex analysis includes:

• evaluation of ordinary and partial differential equations,
• fluid dynamics,
• AC circuits,
• quantum mechanics,
• electromagnetism and acoustics via conformal mapping,
• dispersion relations in photonics via Kramer Kronig relation,
• Z transform and indeed all transform techniques which give rise to signal processing and control theory

But I quickly find that the applications mentioned above is more towards an analytical approach to physical problems that uses only tiny subset of techniques and ideas in complex analysis.

For example, one could get away in basic quantum mechanics and ODE by merely knowing the fact $\sqrt{-1} = i$. Whereas one could get away with conformal mapping with a good grasp of sets and mappings in complex plane without delving into any integration techniques. Complex analysis yields tremendous intuition for PDEs and is used to solve the Laplace equation $\triangle F = 0$, but in reality most problems are solved using meshes and numerical techniques. Actually everything mentioned above perhaps with the exception of fluid dynamics you can get away without knowing much of complex analysis.

Yet every physics and engineering students must take at least one semester of complex analysis which maybe quickly forgotten due to disuse. For example, engineers seldom invoke residue theory when computing the residue for inverting the laplace transform, favoring more heuristic approach such as "the covering" method. Indeed, inverse fourier and laplace transform via Bromwich integral (a complex integral) is never taught in engineering. Where although complex analysis has tremendous usage in quantum field theory and string theory, the "application" is towards theoretical calculations and proofs such as hand computation of green's function for propagators with emphasis on "hand".

• Can anyone suggest some directions for using complex analysis in ways that goes beyond doing mostly one shot, classical physics problems by hand?
• Are there any references out there that tries to apply results in complex analysis in a more computational oriented setting?
• If not, I will be happy to know if there are other areas of complex analysis not covered in the list I have given above and please provide some references to for me look over those applications.
• Actually, not all engineers have a semester of complex analysis; some only get to see complex numbers.
– J W
Nov 28 '20 at 19:28

## 2 Answers

In computer science, specifically in combinatorics (much used in algorithm analysis) one important task is to derive asymptotic behaviour of sequences, which in turn are easiest to get in form of a generating function. The techniques used are heavily based on complex analysis. See for example Flajolet and Sedgewick's "Analytic Combinatorics" (Cambridge University Press, 2009).

• Also the nice book "generatingfunctionology" is online, google it! Sep 12 '15 at 15:47

Finally there is a book

https://www.crcpress.com/A-MatLab-Companion-to-Complex-Variables/Wunsch/p/book/9781498755672

"MATLAB companion to complex variables" published Aug 2016. I can't wait till my library acquire this book.