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Real Analysis vs. Complex Analysis

Date: 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   -----------

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

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
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Associated Topics:
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