Erdős' Anti-Calculus Proposition

Proposition

Erdös contends that in Complex Analysis, versus Single Variable Calculus, a function will not have a derivative of zero when it reaches a maximum value.

In Single Variable Calculus:

A function \(f(x)\), which is smooth (continuous and differentiable) on a open domain \(D\), will have a derivative of zero when it attains a maximum on \(D\).

However, for a complex function \(g(z)\), on a closed domain \(U\) with border \(B\), the maximum (Guaranteed by the Extreme Value Theorem), may not have a derivative of zero.

Proof

Let there exist a complex \(\epsilon\) such that \(|\epsilon| > 0\) and \(\delta\) such that \(|\delta| > 0\) where \(\epsilon\) and \(\delta\) are arbitrarily small.

Furthermore, let the complex function \(f(z)\) be analytic and non-constant on an open domain \(D\), bounded by a contour \(B\) along which \(f(z)\) is continuous and differentiable. Thus, \(f\) is continuous and differentiable on the closed region \(U = D \cup B\).

By the Maximum Modulus Theorem, the maximum of \(f\) occurs on \(B\) at \(z_0\).

Now, suppose that \(f'(z_0)\) is indeed zero. This would mean that for some \(\epsilon\), there is are points \(w = z_0 + \delta\) where \(\delta < \epsilon\) for which \(f(w)=f(z_0)\), since \(f(z)\) is continuous. \(z_0 + \delta\) is a maximum point for \(f(z)\) on \(D\). By the Maximum Modulus Theorem, this means that \(f\) must be constant on \(D\).

Since \(f(z)\) is non-constant on \(D\), this is a contradiction. \(#\)

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.

Present Tense Verbs, Their Meanings and Their Conjugations

Verb Conjugations

A note on pronunciation:
For the most part, all consonants are pronounced as they would be in English, with one notable exception: v is the Bilabial voiced fricative, rather than the labial voiced fricative it is in English.

The vowels are as follows:

• a: the ah sound in far
• e: the eh sound them
• i: the ee sound in deem
• o: the o sound in bone
• u: the oo sound in boon
• ä: the ay sound in hay
• á,é,í,ó,ú,'ä: This indicates that there is a stop before the beginning of the vowel. It is used when there are two vowels right next to each other, but the second is in a separate syllable from the first.

Overview of Sustätyes

Sustätyes is the language of the ancestor peoples to the current people of the Kingdom of Fortuna. The city of Blackport, currently the capital of the nation, is the site of their old City-state, Voltunwe (meaning the Life of Fate). These people were often attacked by raiders and people from other nations, and needed to work together in order to keep strong. Legend has it that these people, the Voltunsus (Fated People), living in Voltunwe, suffered from a period of much violence within themselves that nearly spelled their doom. To combat this, they developed an extremely sophisticated--and since lost to the ages--system of justice. Apparently this system quickly dispelled most, if not all such activity very quickly, allowing the focus to return to the pressing matters of foreign raiders.

As a result of this Justice system, the dichotomy between truth and falsehood became a very important concept to the Voltunsus. Telling a malicious truth was bad, but a malicious lie was worse, a comforting lie wasn't all that bad a thing to do, but if one could tell the whole truth and still be comforting, that was even better. Such comparisons are at the heart of Voltunsus culture, and therefore their language.

Having been continually used for legal purposes since the advent of the language until the present, much is known about the language, even (unlike the Latin of Earth and the Roman Empire) the phonetics and pronunciation rules survive until this day.

Less is known, however, about the daily activities and culture of the Voltunsus. For example, it is known that they were monotheistic in their primary faith, but the nature of their god is unknown. It could have been a many-faced deity just as easily as one with innumerable servants that ran different aspects of the world or a god that resembled Earth's Abrahamic deity of Christianity, Judaism, and Islam. About five hundred years ago, a new faith emerged, through some combination of evangelicalism and revelation, resulting in the destruction of the old, now heretical, texts.

Also, it can be assumed, as the city of Voltunwe sat on the coast, with extensive waterfront access and a very useful anchorage, that a good deal of the City-State's economy came from marine occupations, but whether that was more fishing than trade, or vice versa, is almost entirely guesswork by modern historians and archaeologists.

Hello and About Me (Hallo und Über Mich)

Hello Anybody and Everybody!
My name is Andrew Schade, and this is my blog. Here I will write about my multiple hobbies, my philosophy and other things that I cannot bear to keep silent about.

I am (as of Sept. 2013) a Senior in High School at the Maine School of Science and Mathematics, which is a residential, magnet school in Limestone, Maine, USA and am studying a number of topics, including Computational Physics, Complex Analysis, Human Anatomy, and British Literature. Besides studying, I spend my time working on the novel I am writing, Princess of the Insane, constructing a Conlang,called Sustätyes, programming, and generally being introspective.

I will post news about Princess of the Insane and Sustätyes to this blog (which is going to be its primary function). Additionally, if I feel the urge, I will write articles about whatever issue or topic I feel like ranting about.