# Intro to complex variables

1. May 2, 2006

### axeae

I'm wondering if anyone can recommend a good intro to complex variables book for an undergrad (calc1-3,diffeq). Preferably something that can be found cheap on amazon (Dover?)

2. May 2, 2006

### Tom Mattson

Staff Emeritus
I like the Schaum's outline myself.

3. May 2, 2006

### Staff: Mentor

There's a good short into in Appendix B of "Designing Digital Filters" by Charles Williams. It's a very good intro book to DSP. You can probably just check it out of your technical library, rather than having to buy the book. If you're interested in DSP at all, though, it's a good book to buy.

4. May 7, 2006

### axeae

Thanks for the suggestions. Tom Mattson, you were my Calculus I teacher. I have a few Shaum's Outlines but I was looking for something closer to an actual text book since I'm planning on self-studying.

Last edited: May 7, 2006
5. May 8, 2006

### mathwonk

the all time classic cheap intro, and excdllent, is by konrad knopp.

theory of functions, vols 1 and 2 are bound together in this paperback for $11. Product Details: ISBN: 0486692191 Format: Paperback, 320pp Pub. Date: August 1996 Publisher: Dover Publications the book by richard silverman is also good, based on work of markushevich. the remarkable thing abiout compelx anaklysis is it is such a classical and beautiful subject, it is ahrd for anyone to change it much, hence ahrd for anyone to mess it up, so almost all books are excellent. thus it makes little sense to pay a high price for a book of this nature. I happen also to like Lang's book at 50-60dollars, and there is a similarly priced, more advanced book by eli stein, princeton lectures, that looks impressive. 6. May 8, 2006 ### mathwonk or try this little number: a translation of the original 100 year old treatise by edouard goursat, presumably the man the cauchy goursat theorem is named after. this is a masterpiece. Wonder Book and Video 1306 West Patrick St Frederick MD, 21703 (301) 694-5955 Functions of a Complex Variable Goursat, Edouard US$ 7.19

7. May 8, 2006

It is indeed the same Goursat.

8. May 8, 2006

### Cincinnatus

I'm taking the class now, I really dislike the book the professor assigned for the class. I've been reading the book by Serge Lang instead and liking it much better.

9. May 8, 2006

### mathwonk

there are also several free sets of notes on the web.

10. May 10, 2006

### Tom Mattson

Staff Emeritus
Well howdy! Was it last summer?

When I took complex variables the textbook was the classic by Churchill and Brown. I found it tough to read at the time so I supplemented it with the Schaum's outline, and I got along just fine. The great thing about the outline is that there are a ton of solved problems in it. And of course, you now know about Physics Forums (aka, The Best Damn Website On The Internet, Period.)

11. May 10, 2006

### eok20

The schaum's outline was the only text used in my complex variables course last semester and it's pretty good. Also, I just picked up Visual Complex Analysis by Needham and it is very good. It is not rigorous but really gives you a feel for the material and it has many interesting problems.

12. May 10, 2006

### mathwonk

my favorite is probably the book I taught from last time, and which John Tate used at Harvard in the 1960's, namely Elementary theory of functions of one and several complex variables, by henri cartan, now available as a cheap ($12.55 on B&N) dover paperback. i highly recommend it. The book by churchill is an old, not too rigorous, classic, but now sells for over$100 dollars new, which is ludicrous for math types as it is nowhere near in the league with cartan, not worth a fraction. of course used copies probably abound. i have also taught from churchhill, several decades ago, in the early 1970's, and found it amusing and instructive in the way the old thomas calculus book was instructive. i.e. nothing deep, but lots of practical hands - on engineering type problems and explanations.

i.e. you learn something, but not much theory. as i was reading through thomas in 1970 to prepare to teach from it, i remember saying to myself, "how interesting, a math book with no theorems. but gee i didn't know this example, or this one, and they are nice and illustrative."

you might read churchill for practical information and then cartan to find out what is going on. another great classic is the two volume work by einar hille. its the only book i have with the proof of the big picard theorem.

applied complex variables is good too by john dettman, for $17, paper. one of my outstanding high school students bought it and read it on his own and recommended it to me. francis flanigan is a good authoir and has a cheap paperback too. the book by gareth jones, and singerman is kind of fun too. eberhard freitag is an outstanding complex analyst too with a recent book for about$50, but given the choices above for under $20, from cartan, to goursat, to dettman, to knopp, it seems that paying$100 or more for churchill's or jerry marsden's book would surely be foolish.

if you want to pay $100 or so, the two volume set by r. narasimhan would be outstanding (one and several complex variables). reinhold remmerts book includes historical sources and their translations, seems potentially wonderful. as a second book, i recommend algebraic curves and riemann surfaces, by rick miranda, a wonderful book for$45.

Last edited: May 10, 2006
13. May 10, 2006

### mathwonk

by the way, there are a lot of calculus books by einar hille out there on the used market for $1, yes thats right, one dollar. given that fact, anyone who buys a copy of stewart, or leithold, or thomas - finney, or whatever for$100, is a prime sucker.

oh my word, hille's analytic function theory is available used on the web for as little as $5 for vol I, and$25 for volume II. oh my gawd. jump on it. at least vol 1. and cartan. and goursat.

Last edited: May 10, 2006
14. May 10, 2006

### mathwonk

free intro to complex variables part 1

First Day: What is complex analysis , and why is it interesting?

Complex analysis is the study of the (differentiable) solutions of a single very important differential equation, the Cauchy-Riemann differential equation: ∂f/∂zbar = 0, where ∂f/∂zbar = (1/2)[∂f/∂x+i∂f/∂y]. We can explain this a bit more as follows: Recall that two functions f,g:U-->R^2 are called tangent at the point p of the open set U in R^2 if:
1) f(p)=g(p), and
2) (as z--> p), ( |f(z)-g(z)|/|z-p| --> 0 ).

(It means their graphs are tangent at [p,f(p)] - think about it.) A function L:R^2-->R^2 is called R-linear if L(z+w)=L(z)+L(w) for all z,w in R^2, and if L(tz)=tL(z) for all t in R and all z in R^2. Then a function f:U-->R^2 is called (R-) differentiable at a point p in the open set U, if there is some R-linear function L such that f(z) is tangent at p to the function L(z-p)+f(p), (as a function of z). L is called the (Frechet) derivative of f at p. [See Rudin's Principles of Mathematical Analysis defn.9.11, p.212, or Spivak's Calculus on Manifolds, chap.2, pp.15-16 if you don't remember this definition of derivative.]

Now since R^2 has a natural structure of multiplication, making it into the complex numbers C, just by setting [1,0] = 1 and [0,1] = i, and putting i^2 = -1, we may generalize the last definition as follows: a function f:U-->C is said to be complex differentiable, or holomorphic, at the point p of the open set U in C, if f(z) is tangent at p to L(z-p)+f(p) for some C-linear function L:C-->C.

Of course L is called C-linear if L(z+w)= L(z)+L(w) for all z,w in C, and if L(tz)=tL(z) for all t,z in C. The only difference here is that the scalar t in the second condition is allowed to be any complex number and is not restricted to be a real number as in the definition of R-differentiable, but what a difference it turns out to make!

Complex analysis is the study of holomorphic functions.

Exercise: If f:U-->R^2 is a differentiable function at p, with (R-linear) derivative L, where z=x+iy, then ∂f/∂x and ∂f/∂y are both defined at p, and L is the linear functionwith matrix [ ∂f/∂x ∂f/∂y ], where both entries are regarded as column vectors. Moreover L is actually C-linear (i.e. f is holomorphic) if and only if ∂f/∂zbar = 0.

Thus our two conditions for holomorphicity are equivalent. [At least they are equivalent for differentiable functions. It is conceivable however that a function f exists which has partial derivatives ∂f/∂x and ∂f/∂y which satisfy the equation ∂f/∂zbar = 0, but without f even being continuous, much less differentiable, and thus not holomorphic.

Recall that a function can have partial derivatives without having a (total) derivative. There do exist functions having these properties at least at one point, but I don't know whether any f exists which has partials that satisfy the Cauchy-Riemann differential equation on a whole open set without f being continous on that set. The Looman-Menchoff theorem, p. 43, Narasimhan's Complex Analysis in One Variable, states that if a continuous function on an open set has partials everywhere on that open set which satisfy the Cauchy-Riemann equation then the function is holomorphic.]

Now why would anyone want to study holomorphic functions? [Life is too short to study them simply because they may occur on the Ph.D preliminary exams.] The study of complex numbers arose in the 19th century out of the study of real analysis and algebra, when it was found that the purely formal use of complex numbers actually facilitated the solution of some problems which appeared to concern only real numbers.

There is a nice discussion of this in the book of Osgood. Once complex numbers were accepted as inevitable, which they must be if one ever wishes to solve simple algebraic equations like x^2+1=0, it becomes natural to ask about the calculus of complex numbers. From the point of view of differential equations holomorphic functions turn out also to be closely linked to real harmonic functions, and these had been discovered to be important in physics. That is, the function u(x,y) defining the temperature at a variable point inside a metal disc when that temperature is constant in time, can be shown from simple physical assumptions, to satisfy the Laplace equation ∂^2u/∂x^2 + ∂^2u/∂y^2=0.

Now it is not hard to check that whenever f is holomorphic and has continuous partials of first and second order, then both the real and imaginary parts u,v of f=u+iv satisfy the Laplace equation, i.e. are harmonic. Moreover the relation between u and v is one which has physical significance. So our interest in physics, heat and electricity and magnetism, could lead us to study complex analysis.

Complex analysis is also related to the science of map making! That is, the functions f:R^2-->R^2 which preserve angles locally (i.e. in very small regions), the so-called conformal mappings, are very closely akin to holomorphic functions, indeed they are almost the same thing.

In mathematics, holomorphic functions help us understand questions concerning the radius of convergence of a power series. For example, the real differentiable function f(t)=1/(1+t^2), has infinitely many derivatives at every point of the real line, but its Taylor series centered at the point t=0 only converges on an interval of radius 1. Why? It turns out that the radius of convergence is determined by the phenomenon of absolute convergence, and hence the series cannot converge at any real point whose absolute value is greater than the absolute value of a complex number for which it does not converge.

In this example the series fails to converge at z=i, and hence cannot converge at any real number of absolute value greater than | i | = 1. I am told that such questions have applications, within numerical analysis, to giving limitations on the domain of validity of certain numerical approximations. As another example, one not only needs complex numbers to solve for the eigenvalues of matrices in finite dimensions, but one uses complex path integration to study the analogue of the set of eigenvalues, the spectrum, for complex operators in infinite dimensions, [see Lorch, Spectral Theory, chap.4]. Further, the practice of substituting linear transformations into polynomials as in the Cayley-Hamilton theorem, is mirrored by the study of holomorphic functions of complex linear operators in functional analysis.

Complex analysis and the associated study of the topology of Riemann surfaces also sheds light on the historically interesting question of studying the "non-elementary" transcendental (e.g. elliptic) functions defined by integrating differentials like dt/(p(t))1/2, where p is a polynomial of degree at least three.

This gives some idea of some problems that might push one to study complex analysis, and which indeed did so to the great mathematicians of the previous century, but with the benefit of hindsight we can now look back on their successes and find further motivation for us to study such functions, even after the original motivating problems which led to their investigation may have been settled. That is to say, today the hypothesis of holomorphicity is one which is known to have tremendous mathematical consequences, and thus to take advantage of them in solving our own problems, it is useful to us to be able to recognize holomorphic functions when we meet them, and to know what properties they are guaranteed to have.

15. May 10, 2006

### mathwonk

free intro to complex variables part 2

In many ways the property of holomorphicity is analogous to that of being a polynomial, but more subtle. For example one is always interested, in mathematics, in the problem of existence and uniqueness of solutions to equations. For polynomials the fundamental theorem of algebra tells us that if f(z) is a complex polynomial, and if z0 is a complex number such that f(z) = z0 has no solutions, then f is a constant.

The analogous existence theorem is this: if f:C-->C is a holomorphic function (defined and holomorphic everywhere), and if there are two complex numbers z0, and z1, such that neither of the equations f(z)= z0, nor f(z)= z1 has a solution, then f is a constant.

The corresponding uniqueness theorems are these: if f,g are complex polynomials with exactly the same zeroes (including multiplicities), then the quotient f/g is a non zero constant; and if f,g:C-->C are holomorphic functions with exactly the same zeroes, then f/g=e^h , for some holomorphic function h. Since e^z is never zero, even for complex z, this is the best we could hope for.

There are other analogies with polynomials, as follows: if one is given a finite set {a1,...,an} of complex numbers, not necessarily distinct, then there exists a polynomial f(z) of degree n, whose set of zeroes is precisely this set, and such an f can be expressed as a (finite) product, one factor for each ai. Moreover, if f is a complex polynomial of degree n, and if f vanishes (i.e. has the value zero) at a set of n+1 distinct points, then f is the zero polynomial. Hence a polynomial of degree n is entirely known if its values are known on a given set of n+1 points.

Moreover there is a formula, called Lagrange's interpolation formula, which expresses the value at any point in terms of a finite sum involving the values at the given n+1 points.

[The generalization to holomorphic functions is perhaps not so obvious - can you guess it? For the second part you have to figure out what we would mean by a set of infinity plus one points!]

In fact if {a1,...,an,......} is an infinite set of complex numbers, such that in any circle centered at the origin there are only a finite number of them, then there is a holomorphic function f:C-->C whose zeroes occur precisely at the given points (and with given multiplicities), and this holomorphic function can be expressed as an infinite product, with essentially one factor for each point.

Moreover, if {a1,...,an,......} is a infinite sequence of distinct complex numbers which converges to a (finite) complex number a0, and if f:C-->C is an entire holomorphic function, (holomorphic functions defined on all of C are called entire), which vanishes on all the points ai in the sequence and hence also at a0, then f is identically zero.

Consequently an entire holomorphic function is completely determined by its values on the points of a convergent sequence. So the notion of "infinity plus one" points turns out to mean an infinite sequence plus a limit point of that sequence. In particular, an entire holomorphic function is determined by its values on any circle, and indeed there is a formula, Cauchy's integral formula, which gives the values of the function at points inside the circle in terms of an integral over the values on the circumference of the circle!

In fact an entire holomorphic function is always defined by an infinite polynomial, i.e. a convergent power series.

Last edited: May 10, 2006
16. May 10, 2006

### mathwonk

free intro to cx vbls, part 3

The uniqueness property just stated tells us in particular that a differentiable function f:R-->R has at most one extension to a holomorphic function F:C-->C, whereas f always has infinitely many different extensions to a (R-) differentiable function F:R^2-->R^2. For example, sin(t), e^t, and a0+a1t+a2t^2+....+ant^n, each has a unique extension to a holomorphic function on C, but 1/(1+t^2), has no extension to a holomorphic function on all of C, and f(t)=exp(-1/t^2) for t≠0 and f(0)=0, has no extension which is holomorphic on any neighborhood of 0, even though f(t) is infinitely differentiable on R.

Now the function 1/(1+t^2) does of course have a holomorphic extension to any open set in C that does not contain either of the points i or -i, and the extension is given by the same formula, 1/(1+z^2). On the other hand there exist functions f:R-->R which have holomorphic extensions to certain open subsets of C, but such that the extensions may take several different values at the same point of C! That is, given a point z0 in C, there may be one holomorphic extension of f to one open set U containing z0 which has one value at z0, and there may be a second holomorphic extension of f to a second open set V which also contains z0 such that the second extension has a different value at z0 from the first one!

For example, the real function f(t)=ln(t), defined and differentiable for t in R+ can be extended to a holomorphic function on the complement of any ray emanating from the origin in the complex plane. But if F1 denotes the holomorphic extension to the complement of the negative imaginary (y-) axis, and if F2 denotes the holomorphic extension to the complement of the positive imaginary axis, then F1(-1)= iπ, while F2(-1)= -iπ.

in fact complex variables reveals that the two examples above, 1/t and 1/(1+t^2) are not so different.

integration of 1/z along paths that miss zero (and infinity) is analogous to integration of 1/(1+z^2) along paths that miss either i or -i.

thus there is a cionnection between log(z) and arctan(z). they are essentially the same under the transformation (z-i)/(z+i) that sends i and -i to zero and infinity!

So in cases of functions which extend only into part but not all of C, the values of the extension at a specific point are not determined by the point but rather by the whole open set on which the extension exists. It is true that for each connected open subset of C which meets R, there is at most one holomorphic extension of f to that set. So the holomorphic extension of a function is often inevitably multiple valued, and the attempt to work out the proper domain on which a full single-valued holomorphic extension can be defined led Riemann to the beautiful theory now called "Riemann surfaces".

Riemann analyzed the geometry of the resulting domains and thereby constructed the origins of the subject of topology. This problem was also studied abstractly by Abel and Galois in connection with the problem of integrating algebraic functions. [An algebraic function is one g(z) such that there exists a polynomial F(z,w) in two variables for which F(z,g(z))=0, for all z. g(z)=z^1/2 is such a function since it satisfies this condition for F(z,w)=z-w^2.] Indeed, Galois wrote down some of his very advanced insights on this question in the second part of the same famous letter which is so much better known for its statement of the criterion for a polynomial equation to be solvable in terms of radicals.

So much for motivation. Don't be dismayed if you don't follow all of this. Read it and think about it and refer back to it when the course has progressed further, and see how much more you will understand of it.

17. May 10, 2006

### Cincinnatus

That was a great series of posts Mathwonk.

I just finished my first complex analysis class and hated every minute of it.
The way my class was taught it seemed like the subject was just a bunch of cookbook like solutions to different example problems.

But after reading a good portion of Serge Lang's book, and now your posts I'm much more interested. I wish my class hadn't been taught the way it was...

18. May 10, 2006

### mathwonk

you have no idea how happy i am for your kind feedback. often i feel as if i am talking to myself. I put a lot of energy and time and thought into those writings and I am delighted if they help or motivate anyone. best regards.

those posts formed the introductory lecture to the course i taught in 1989 in grad complex.

I have a lot more, but not a complete set of notes from that course.

Last edited: May 10, 2006
19. May 11, 2006

### axeae

Wow, great post. I've been a little busy with finals so I haven't had a chance to check out the thread but its been a great help, I'm actually going to print it out now. I should be ordering quite a few things off of Amazon this weekend.

Tom Mattson, it was last summer, session II. Your calculus class actually made me want to be a math major, so I've been doing math nonstop since.

Last edited: May 11, 2006
20. May 11, 2006

### Tom Mattson

Staff Emeritus