# Lower-division complex analysis textbook

I'm looking for recommendations for a good textbook to use for a hypothetical lower-division course in complex analysis, at a level of sophistication comparable to a second or third semester course in calculus. (In particular, the audience are not necessarily math majors, and the course would not be proof-based.) I'm particularly interested in books that have a good collection of exercises at this level: hopefully a good number of basic computational exercises, plus some more conceptual but not overly difficult (and in particular not proof-based) ones.

• Who is hypothetically taking this course and why? What previous math have they had? Does it satisfy a requirement for them? – Ben Crowell Apr 16 '19 at 2:37
• I don't know of a book that's substantially below the level of standard undergraduate texts such as those by Richard A. Silverman, Ruel V. Churchhill, Jerold E. Marsden, etc. but in the late 1990s I incorporated in to a high-honors level post-BC honors level calculus class a fair amount of material that one finds in such books prior to the introduction of complex integration, such as connections between trig functions and hyperbolic functions, Cauchy-Riemann equations (checking they hold for various elementary functions using De Moivre's theorem makes for some interesting (continued) – Dave L Renfro Apr 16 '19 at 17:29
• algebra workouts), various results about roots of unity such as solving by quadratics $x^n = 1$ for $n \leq 6$ and $n=8,$ correspondence between $\mathbb C$ and certain $2 \times 2$ real matrices, values of $i^i$ and related things with Euler's formula, ways of formally carrying out certain types of integrations using complex exponentials (e.g. the integral of $e^{3x}\sin 2x),$ and perhaps some other things I've forgotten about now. I assembled the material from many books and papers (e.g. this and this). – Dave L Renfro Apr 16 '19 at 17:46
• I think you should at least look at Saff and Snider's text. It has a nice writing style and is quite computational with some rather intuitive commentary to understand why theorems are true. It is more like a calculus text than a proof text. – James S. Cook Apr 17 '19 at 0:19
• Incidentally, two topics I considered including but didn't (because of time constraints) that you definitely want to consider are elementary issues involving linear fractional transformations and some basic conformal mapping tasks (mapping various regions onto other regions). Again, I don't know of a book for the level I'm thinking about, but it certainly seems something that could fill a void if anyone were sufficiently interested in writing such a book. I'm thinking of a complex variable analog of how U.S. 1st/2nd year linear algebra texts are relative to their upper level versions. – Dave L Renfro Apr 17 '19 at 8:01

I really liked using Zill and Shanahan "A First Course in Complex Analysis with Applications" when I taught this course, certainly at a low level of sophistication. Loads of pictures, nice wide margins, careful with algebra saved "for the reader" in other texts. I had a version from 2003 in softcover, not sure if that is available now. I also find it prepared those with better uptake for what they might encounter next.

On a note I cannot vouch for from personal use, there is a text (same publisher!) by Howell and Mathews which is closely connected to several pedagogical initiatives surrounding complex analysis, for further details on which see for instance the contents of this special issue of PRIMUS. Not that you have to read or use those, but just for information as to where other people are no doubt teaching this same cohort and trying to share best practices.

Take a look at the Schaum's Outline of Complex Variables Theory and Practice. I find it friendly in being very problem based. Warning, it does have SOME proofs but not a huge amount. Or "proofs" that are smaller, easier little derivations. And it has a gazillion (technical term) calculational problems.

When you look at it, you can see if you like or dislike the review text style. There's less text than normal in a doorstopper book but at the same time it's not "hard" like some very terse short that is more monograph style (e.g. Adrian Albert Higher Algebra). More short and practical like a language review grammar is economical (which I prefer to standard language texts). I personally like that but see how you feel.

The level is pretty friendly in terms of not expecting students to have had real analysis or "advanced calculus" first. Just starts with the standard review of complex alebra, doesn't get into a bunch of series convergence assumptions (or expect students to know them for homework problems) at least not in the beginning or without developing needed tools. For comparison, it's at an easier level than Carrier Krook Pearson where first chapter had HW problems that blithely ask for proofs in the well known theory of real variables (not well known to kids with basic calculational calculus).

Caveat: I have the first edition. But I suspect the second edition is fine. But just letting you know I haven't looked at it.

Price: \$14 new, for second edition, paperback.

• Ugh, Schaum's is awful. Readers here should be aware that "guest" is committed to a no-explanations curriculum, and this is what you get with Schaum's. – Daniel R. Collins Apr 16 '19 at 3:25
• True regarding "guest", but Schaum's could be useful to supplement a course with teacher-supplied explanations with lots of exercises - I have done so effectively in a real analysis course. You just can't use it as "the textbook". – kcrisman Apr 18 '19 at 12:16