Real Analysis vs. Complex AnalysisDate: 11/15/97 at 01:38:59 From: Tony Asdourian Subject: Real Analysis vs. Complex Analysis Dear Dr. Math, In teaching complex numbers to my students, the subject of real analysis and complex analysis came up. I mentioned that both are courses taught at university, and my students wanted to know why one had to study real analysis at all if one could study complex analysis, since they felt one could just study any question in real analysis with the tools of complex anlaysis by assuming the imaginary part 0. I gave a two-part reply: 1) Perhaps one must study Real Analysis first to study Complex analysis; that is, rigorous definitions of limits and continuity and derivatives must be understood clearly with real numbers before one can understand analagous definitions with complex numbers. 2) Perhaps the very fact that real analysis deals with a restriction of no imaginary part gives it some special properties that are not true with complex numbers, and thus one would get different results based on that fact. The analogy I gave was the study of linear equations in general vs. Diophantine equations, where the very fact that Diophantine equations are restricted to integers gives them special and interesting properties that linear equations that can take on all real values do not have. Are these explanations correct? Any clarification you could bring to as to why one studies complex analysis AND real analysis, instead of just complex analysis, would be appreciated. Tony Asdourian Date: 11/17/97 at 19:14:33 From: Doctor Tom Subject: Re: Real Analysis vs. Complex Analysis Hi Tony, The name "complex analysis" is a little misleading, since the subject in reality investigates only those functions of complex numbers that have a derivative. The idea of a derivative in the complex plane looks superficially the same as a derivative on the real line: A real function f has a derivative at x if: limit f(x+h)-f(x) h->0 ----------- h exists. For complex derivatives, it looks exactly the same, except that we usually write "z" for "x" to remind ourselves that f is a function of a complex variable. The key difference is that if h is real, it can only approach zero from above or below. If h is complex, it can approach zero not only from an infinite number of directions, but it can spiral in, etc. Thus differentiability in the complex plane depends on the existence of a vastly more restrictive limit. Almost every (in a measure-theoretic sense) differentiable real function is not differentiable in the complex sense. If you stick to the "standard" functions, it looks like the list is similar, but that's highly misleading. But you can construct examples using "standard" functions that blow up too. For example, consider the function: f(x) = e^(-1/x^2), if x is not zero, and f(0) = 0. On the real line, as x approaches zero from either direction, f(x) is like e raised to the power of "minus infinity" - it goes to zero extremely rapidly (and in fact, is infinitely differentiable). On the other hand, in the complex plane, replace the "x" with a "z" and let the z values tend to zero along the imaginary axis. The function tends to e to the power of positive infinity - the function isn't even continuous, let alone differentiable. The condition of differentiability is so strong in the complex plane that if a function has one derivative, it has all derivatives. This is certainly not true of the real functions. This strong condition is so fascinating that the whole subject of complex analysis is basically the study of infinitely differentiable functions on regions of the complex plane. There's also an interlocking of the real and imaginary parts of the range of a complex funtion that's totally different from, say, a study of pairs of real functions of two variables. There is an overlap, called "harmonic analysis," which uses real methods to study harmonic functions which are solutions to a certain type of differential equation, and whose properties are similar to differentiable functions of a complex variable. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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