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In complex analysis, we care about the functions of a complex variable. Are such functions just special case of the complex valued function defined in Rudin's Principles of Mathematical Analysis, since they have many similar properties such as uniformly convergence in series. Moreover, I wonder the relationship between the functions of two real variables and the functions of a complex variable.

Thanks

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For the last part, you should see the following if you haven't yet, to see whether your question is already answered there: math.stackexchange.com/questions/5108/… –  Jonas Meyer Nov 12 '10 at 23:49
    
For the first part, I'm not sure what you're asking. Yes they are a special case of complex-valued functions, and if you're talking about holomorphic (a.k.a. complex analytic) functions, they are more specially a case of continuous complex-valued functions, and even more specially a case of differentiable functions on open subsets of the plane. (They even turn out to be infinitely differentiable, and to have power series representations about each point, which is why they're analytic.) –  Jonas Meyer Nov 12 '10 at 23:53
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(As an aside: complex analysis is not really about the study of "special cases of functions of a complex variable", but about the topology of the Riemann sphere with points removed...) –  Willie Wong Nov 13 '10 at 0:09
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@Willie Wong: I would say that there are much more than one reason to study complex analysis.. It really is one of the most beautiful and satisfactory fields among all mathematical fields!! –  AD. Nov 13 '10 at 5:00

4 Answers 4

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$\newcommand{\Hol}{\mathcal{H}}$ Complex analysis (usually) refers specifically to questions involving differentiable functions starting with the deceptively simply looking case of functions of one varible on open subsets of the complex plane. Already here there are many differences from the theory of differentiable functions $f: (a,b) \to \mathbb{R}$. Some examples:

  • If $G \subset \mathbb{C}$ is open and $f: G \to \mathbb{C}$ is once differentiable or has an antiderivative, it is infinitely often differentiable. Such functions are called "holomorphic" or "holomorphic on $G$" if one wants to be precise (sometimes written $f \in \Hol(G)$). This is in stark contrast to the real case where a differentiable function may fail to have even a continuous derivative.
  • If $f \in \Hol(G)$, then for any $z \in G$ there exists an open ball around $z$ contained in $G$ on which $f$ is equal to its own Taylor series at $z$. Even smooth real functions are not this nice. Wikipedia has an article on the standard example - though I also like the Fabius function.
  • Cauchy's integral formula which roughly states that if you know the values of a holomorphic function on the boundary of a disk, you know its values everywhere inside the disk.
  • Liouville's theorem which states that if $f \in \Hol(\mathbb{C})$ is bounded, then $f$ is constant.

Essentially: When you're wondering whether the theory of holomorphic functions is "nice" in some way, the answer is usually yes. Whereas the theory of even smooth functions is a lot more messy.

I have not provided links to Wikipedia's articles on the relevant theorems. Instead I recommend these free notes by Christian Berg from the University of Copenhagen. They are in English and have been in use in the university's first course in complex analysis for close to a decade (to the best of my knowledge) and have had multiple updates and corrections over the years.

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The recommended notes looks like a good introduction. –  AD. Nov 13 '10 at 4:57

For the second part of your question, one can think of complex functions as continuous functions of two real variables that satisfy some differential equations. The differential equations are the Cauchy-Riemann equations mentioned in the answer Jonas Meyer linked to: How is $\mathbb{C}$ different than $\mathbb{R}^2$?

This gives straight away a reason why all functions of two real variables are not functions of a single complex variable, since not all functions of two real variables satisfy those differential equations.

A somewhat deeper understanding of what is going on can be obtained by looking at the complex coordinates themselves. In two real variables we often denote our coordinates by $(x,y)$. Let's agree to identify the complex plane $\mathbb C$ with $\mathbb R^2$ in the usual way (i.e. via $z = x + iy$). Then we can perform a coordinate change; instead of looking at $(x,y)$, we will consider the coordinates $(z,\bar z)$.

This is a very subtle change in point of view, and merits your attention. Please take the time to convince yourself that this amounts to a change of basis of the $\mathbb R$-vector space $\mathbb R^2$: if $e_1 = (1,0)$ and $e_2 = (0,1)$ is the usual basis, then the new basis is given by $e_1 + e_2 = (1,1)$ and $e_1 - e_2 = (1,-1)$.

What this means is that any function $f$ of the real variables $(x,y)$ can be expressed as a function of the new variables $(z,\bar z)$. Now one can say that just as in the case of real analysis we start by considering the case of one real variable, then we want to consider one complex variable to begin with in the case of complex analysis. This leads us to consider the functions $f$ which only depend on the variable $z$ (or $\bar z$, the choice amounting to a choice of a square root of $-1$).

In other words, we consider those functions $f$ of the variables $(z,\bar z)$ which do not depend on $\bar z$. Another way to express this property is to say that we require $f$ to satisfy the differential equation $\partial f/\partial \bar z = 0$, but once you write out what this means in the original coordinates $(x,y)$ this equation turns out to be equivalent to the Cauchy-Riemann equations.

Thus, when we investigate what it means for a function to depend on a single complex variable, we stumble upon the Cauchy-Riemann equations, and thus upon the definition of a holomorphic function.

I wholeheartedly recommend the first chapter of Greene and Krantz's "Function theory of one complex variable" for a more in depth discussion of this point of view. They take some lovely examples with polynomials in $(x,y)$ and then in $z$ which show quite explicitly what is going on.

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also see: en.wikipedia.org/wiki/Split-complex_number –  deoxygerbe Nov 25 '10 at 7:55

Holomorphicity is a stronger condition compared to differentiability in real variables. Here are some theorems of complex analysis, which are not true in real analysis.

  1. If two holomorphic functions agree on an open set, they have to agree everywhere.

  2. Any bounded function holomorphic on all of $\mathbb C$ is necessarily constant.

  3. A nonconstant holomorphic function can miss at most one value.

You can see from these theorems that one has much more control over a holomorphic function, as opposed to a differentiable function of two real variables.

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I'm Not too sure how to answer the first part of your question. For the second part, a complex variable function can be thought of as a real variable function whose domain and range are both subsets of R^2. This is usually different then those functions you study in multivariable calculus: functions from R^n to R. One noticible difference is that for functions of a single complex variable, there is a total derivaitve which exists if the cauchy riemaan differential equations hold. Real functions on R^2 would have two partial derivatives instead.

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