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 continuous 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.
