14 October, 2013

Some Complex Math--the Relation Between Analyticity and Antidifferentiability

In Complex Analysis, the field of mathematics relying on doing almost all types of regular mathematics, but in the Complex Number plane, as opposed to the Real Number line, the concept of Analyticity, the state of a function having the property of being Analytic, is central.

For a function \(f(x)\) to be analytic in the Real Numbers, it must have some power series representation \(\sum{c_n {(x-x_0)}^n}\). A function satisfies this property if it is infinitely differentiable, meaning that \(f'(x), f''(x),\frac{d^3 f}{d x^3}, ..., \frac{d^\infty f}{d x^\infty}\) all exist, and are continuous. Most continuously differentiable functions are analytic in \(R^1\), but there are some functions that are continuous and differentiable that do not satisfy this property, for example, the non-analytic, yet smooth function:
$$f(x)=\begin{cases}\exp(-1/x)&\text{if }x>0,\\ 0&\text{if }x\le0,\end{cases}$$

The proof of Non-analyticity is here.

In the Complex Field, Analyticity is a slightly different animal. In general, if a function, \(f(z)\) is differentiable on some open domain \(D\), it is analytic on that domain. It can then be proved that if a function \(f(z)\) is analytic, it is infinitely differentiable on the same open domain \(D\).

Now, my quandry is thus:

In contour integration of complex functions, one of the more important concepts is the idea of path-independence, or conservativity. A function \(f\) is said to be conservative iff (if and only if) \(\int_a^b {f(z(t)) dt}\) evaluates to the same value no matter what simple smooth path \(z(t)\) represents, so long as \(a\) and \(b\) are the same.

If a function is conservative, then \(\int_\Gamma f(z) dz\) is simple. First, find the antiderivative of \(f\) (its existence is guaranteed based upon its conservativity), next, find the endpoints of \(\Gamma\), and finally, integrate \(f\) from the beginning of  \(\Gamma\) to the end using the Fundamental Theorem of  Calculus.

$$\int_\Gamma f(z) dz = \int\limits_a ^b f(z) dz = (F(b)-F(a))$$

Now, the guarantee of the existence of an antiderivative for \(f\) is bidirectional, meaning that, if \(f\) is continuous, and it has an antiderivative, it must therefore be conservative. This turns out to be extremely powerful, as the analyticity of \(f\) implies that \(f\) is both continuous and has an antiderivative over the domain of analyticity.

My issue is that while it seems obvious that if \(f\) is infinitely differentiable, there must also be a function, \(g\) for which \(g'=f\), the proof for that statement is very non-obvious for me. I believe the answer is Green's theorem, but it sadly uses a very, very long proof that goes through the classification of all possible types of Analytic functions.

Anyways, I hope that my little rant about analyticity has entertained those whose knowledge of complex calculus is greater than mine, and educated those whose knowledge of complex of Calculus was increased by this.

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