Geometric function theory
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.
Riemann mapping theorem
This is critical mathematical knowledge. To non-math experts, it looks abstract. If you live in US, the Fed uses it to calculate inflation. It effects interest rates, whether those on SS get an increase year to year. It effects business decisions, pay of employees, the value of USD relative to EU, YEN etc. What if Bernanke is wrong? In math, there are two types of errors. 1) Sloppiness or incompetence. 2)Using math to lie so most won't know the truth. 'shadow stats.com' uses the Log fn. The fed says inflation is 1-2%. Shadow stats says it exceeds 10%.
If people want truth, they must jettison math fears or notions they are not smart enough. Math is the backbone of all science,all we use including cell phones, cars, bridges and houses. It got us out of caves, it makes paint for art, etc. Math is infinitely valuable for doing anything practical, while like art, has beauty and eloquence. The following essay is presented as 'math is art' theorem style.
No one should be intimidated by the 'language used'. Every specialty has it's jargon. Mathemeticians have theirs. Their language is easier than most jargon, easier than chinese or arabic. I only had one ez calc course in college. I had to buy a number of calc books to find this fn. I knew fed was lying but did not know what a 'geometric fn' was. I found a partial explanation in a calc book for physics majors. I was determined. Why? Using something as beautiful and critical as math to lie is an abomination, my biggest pet peeve.
This is a beatiful explanation of this fn. Dig in, enjoy.
Let z
0 be a point in a simply-connected region D
1 (D
1≠ ℂ) and D
1 having at least two boundary points. Then there exists a unique analytic function w = f(z) mapping D
1 bijectively into the open unit disk |w|<1 such that f(z
0)=0 and
f ′(z
0)>0.
It should be noted that while Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function.
Elaboration
In the above figure, consider D
1 and D
2 as two simply connected regions different from ℂ. The Riemann mapping theorem provides the existence of w=f(z) mapping D
1 onto the unit disk and existence of w=g(z) mapping D
2 onto the unit disk. Thus g-1
f is a one-one mapping of D
1 onto D
2.
If we can show that g-1
, and consequently the composition, is analytic, we then have a conformal mapping of D
1 onto D
2, proving "any two simply connected regions different from the whole plane ℂ can be mapped conformally onto each other."
Univalent function
Of special interest are those complex functions which are one-to-one. That is, for points z
1, z
2, in a domain D, they share a common value, f(z
1)=f(z
2) only if they are the same point (z
1 = z
2). A function f analytic in a domain D is said to be univalent there if it does not take the same value twice for all pairs of distinct points z
1 and z
2 in D, i.e f(z
1)≠f(z
2) implies z
1≠z
2. Alternate terms in common use are schilicht and simple. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.
References
- Krantz, Steven (2006). Geometric Function Theory: Explorations in Complex Analysis. Springer. ISBN 0-8176-4339-7.
- Noor, K.I. Lecture notes on Introduction to Univalent Functions. CIIT, Islamabad, Pakistan.
 | This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it. |