# Complex analysis (Applied versus pure)

I am studying Electrical Engineering and I want to specialize in signal processing. However, I have to study complex analysis first (I am an undergraduate, so I lack some terminology). In your opinion:

1) What's the difference between studying complex analysis in pure and applied contexts (topics I mean, and how much should I dig down toward purity)?

2) Do I need real analysis, and what topics?

3) Please suggest books for both pure and applied classes of either real or complex analysis.

• I would guess Saff and Snider is the text for you. I think one of the authors is an EE-prof. The computational insight of that text is masterful, conversational and really easy to understand. Oct 4 '14 at 18:15
• Whats causing my pain is the epilson-delta definition and limits if i find something on differentiation and integration level with emphasis on visualization it well be good Oct 4 '14 at 21:12
• Visualization is hard unless you really mean heuristics. There is no direct analog of the graphical method you know from real calculus because it would need $4$-dimensions. However, the technique of studying $w=f(z)$ and how curves in $z$-plane map to $w$-plane is helpful... I'm not sure what you're looking for... Oct 6 '14 at 0:12

First: there seems to be a traditional belief that "pure" math fusses over tiny uninteresting details that "applied" math takes for granted, etc. Sure, we can operate this way, and make "pure math" as irrelevant as we want, or caricaturize it as such. Oppositely, if we try, we can caricaturize "applied math" as slip-shod fuzzy-thinking. :)

As I've ranted elsewhere around here, I think that the reality is not nearly as silly or binary. In particular, complex analysis is almost singular among ideas arising in 19th century mathematics, in that everything seems to work out in the best possible way. In particular, for various technical reasons that are not well-delineated by a low-tech picture of "rigor", essentially all the operations one would want to perform, ... interchange of limits, passing differentiations inside integrals, and so on, ... are legal. The problem is that the legality is not so trivial to demonstrate, and, again, low-tech demonstrations are very often semi-incomprehensible/unnatural.

In particular, the contemporary spirit of "(textbook) real analysis" runs off the road very often, apparently teaching us that we should be afraid to do anything, even the obvious. This is not constructive. The potential hazards that "real analysis" brings to our attention mostly do not occur in complex analysis... but, again, it's not so easy to prove that they don't occur... and low-tech (=undergrad, early grad-level) proofs are often extremely opaque.

I would encourage "pure" and "applied" math people alike to first approach complex analysis in a worry-free, happy-heuristic way, to see all the amazing things that it makes possible. To the extent that one worries about edge cases or technicalities, one will (but it's not so easy to see this in advance, nor by low-tech means) find that the things that might have gone wrong (in a different universe) do not do so here.

So, don't worry about "pure/rigorous" complex analysis (and certainly not about undergrad-level "real analysis", which may inhibit you more than it helps you).

(My own notes on complex analysis http://www.math.umn.edu/~garrett/m/complex/ are aimed at math grad students (without pure/applied distinction) who've not taken any course in complex analysis previously. Those notes do not really touch on "engineering" applications, but do try to illustrate how complex analysis is "the correct" fulfillment of what "calculus" should have been. Not worrying toooo much about "analysis/estimates", because it turns out there's rarely any serious issue. The notes are freely available, anyway.)

This may not be a direct hit, but since you mentioned "emphasis on visualization," may I suggest you investigate Tristan Needham's Visual Complex Analysis. E.g., see this MSE answer:

• you anticipated my next comment, but, I'm not so sure this helps with trouble with $\epsilon$ and $\delta$. Oct 6 '14 at 12:45
• Right, I'm currently reading this and I really like it. Instead of 4D visualizations, the usual idiom is to show 2D -> 2D transformations of the plane. Niggling point: The "applications" (thus far) are in terms of historical physics, astronomy, and geometry (Newton, Ptolemy, etc.). Jun 22 '16 at 1:36
• Another book with an emphasis on visualization is Elias Wegert's Visual Complex Functions: An Introduction with Phase Portraits, well worth taking a look at. Wegert sketches an application to signal processing in Chapter 1, but postpones further applications to a yet-to-be-written Volume 2 (math.stackexchange.com/q/1784972/118539).
– J W
Dec 30 '20 at 10:28

Since "epsilon-delta" has been mentioned a couple of times by the question asker, I just thought I'd add my opinion that the epsilon-delta business should not be seen as a divider between pure and applied math. Understanding how sensitive the outputs of a function are to the inputs must certainly be important in real-world applications. Also, one cannot do analysis of any kind (pure or applied, real or complex) without being able to give bounds on various things (such as errors in approximations), which often requires the same types of skills as one encounters while learning to do epsilon-delta proofs in calculus. If you ever do a calculation like "how much wiggle room can I allow in quantity 1 in order to ensure that quantity 2 stays within an acceptable range", then you are doing an epsilon-delta type calculation.

This sort of addresses your question 2. I would consider epsilon-delta a good topic for everyone, including "applied" people.

I should add that I'm not advocating for a highly technical nitpicky approach to learning complex analysis. I agree with the points in paul garrett's answer (which I will avoid paraphrasing so I don't misrepresent his views.) I just don't think one should look for textbooks by filtering to avoid epsilon-delta altogether.

1) "Pure" complex analysis is filled with theory and proofs, and is more like real analysis. "Engineering" complex analysis is more about problems, and more like calculus. I recommend the latter for you, an engineering student.

2) You need real analysis for the former.

3) A good book for "pure" is Rudin A good book for applied is Kreyszig," Advanced Engineering Mathematics, Chapters 11-17.

• Whats causing my pain is the epilson-delta definition and limits if i find something on differentiation and integration level with emphasis on visualization it well be good Oct 4 '14 at 23:31
• Some might disagree with designation of Rudin as a reasonable reference for complex analysis... at least insofar as he mentions almost nothing from the 19th century that motivated development of the subject, for example. That is, it is exactly an "analyst's view" of complex analysis... which is not universal. Oct 6 '14 at 0:40