Geometric function theory: Difference between revisions

Revision as of 13:26, 25 June 2013

Latest revision as of 15:31, 22 January 2024

(29 intermediate revisions by 26 users not shown)

Line 1:

{{Short description|Study of space and shapes locally given by a convergent power series}}

'''Geometric function theory''' is the study of [[Geometry|geometric]] properties of [[analytic function]]s. A fundamental result in the theory is the [[Riemann mapping theorem]].

==Topics in geometric function theory==

==Riemann mapping theorem==

The following are some of the most important topics in geometric function theory:<ref>Hurwitz-Courant, ''Vorlesunger über allgemeine Funcktionen Theorie'', 1922 (4th ed., appendix by H. Röhrl, vol. 3, ''Grundlehren der mathematischen Wissenschaften''. Springer, 1964.)</ref><ref>MSC classification for 30CXX, Geometric Function Theory, retrieved from http://www.ams.org/msc/msc2010.html on September 16, 2014.</ref>

===Conformal maps===

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

{{main|Conformal map}}

to EU, YEN etc. What if Bernanke is wrong? In math, there are two types of errors. 1) Sloppiness

[[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a conformal map ''f'' (bottom). It is seen that ''f'' maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.]]

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%.

A '''conformal map''' is a [[function (mathematics)|function]] which preserves [[angle]]s locally. In the most common case the function has a [[Domain of a function|___domain]] and [[Range of a function|range]] in the [[complex plane]].

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.

More formally, a map,

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.

: <math>f: U \rightarrow V\qquad</math> with <math>U,V \subset \mathbb{C}^n</math>

This is a beatiful explanation of this fn. Dig in, enjoy.

Let ''z''{{su|b=0}} be a point in a simply-connected region ''D''{{su|b=1}} (''D''{{su|b=1}}≠ ℂ) and ''D''{{su|b=1}} having at least two boundary points. Then there exists a unique analytic function ''w = f(z)'' mapping ''D''{{su|b=1}} bijectively into the open unit disk |''w''|<1 such that ''f(''z''{{su|b=0}})''=0 and

''f ′(''z''{{su|b=0}})''>0.

is called '''conformal''' (or '''angle-preserving''') at a point <math>u_0</math> if it preserves oriented angles between [[curve]]s through <math>u_0</math> with respect to their [[orientation (mathematics)|orientation]] (i.e., not just the magnitude of the angle). Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or [[curvature]].

It should be noted that while [[Riemann's mapping theorem]] demonstrates the existence of a mapping function, it does not actually ''exhibit'' this function.

===Quasiconformal maps===

{{main|Quasiconformal mapping}}

In mathematical [[complex analysis]], a '''quasiconformal mapping''', introduced by {{harvtxt|Grötzsch|1928}} and named by {{harvtxt|Ahlfors|1935}}, is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded [[ellipse#Eccentricity|eccentricity]].

Intuitively, let ''f'' : ''D''&nbsp;→&nbsp;''D''′ be an [[orientation (mathematics)|orientation]]-preserving [[homeomorphism]] between [[open set]]s in the plane. If ''f'' is [[continuously differentiable]], then it is ''K''-quasiconformal if the derivative of ''f'' at every point maps circles to ellipses with eccentricity bounded by ''K''.

If ''K'' is 0, then the function is [[conformal map|conformal]].

===Analytic continuation===

[[Image:Imaginary log analytic continuation.png|316px|right|thumb|Analytic continuation of natural logarithm (imaginary part)]]

'''Analytic continuation''' is a technique to extend the [[___domain of a function|___domain]] of a given [[analytic function]]. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an [[infinite series]] representation in terms of which it is initially defined becomes divergent.

The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of [[mathematical singularities]]. The case of [[several complex variables]] is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of [[sheaf cohomology]].

===Geometric properties of polynomials and algebraic functions===

Topics in this area include Riemann surfaces for algebraic functions and zeros for algebraic functions.

====Riemann surface====

{{main|Riemann surface}}

A '''Riemann surface''', first studied by and named after [[Bernhard Riemann]], is a one-dimensional [[complex manifold]]. Riemann surfaces can be thought of as deformed versions of the [[complex plane]]: locally near every point they look like patches of the complex plane, but the global [[topology]] can be quite different. For example, they can look like a [[sphere]] or a [[torus]] or several sheets glued together.

The main point of Riemann surfaces is that [[holomorphic function]]s may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially [[multi-valued function]]s such as the [[square root]] and other [[algebraic function]]s, or the [[natural logarithm|logarithm]].

===Extremal problems===

Topics in this area include "Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations".<ref>MSC80 in the MSC classification system</ref>

===Univalent and multivalent functions===

{{main|Univalent function}}

A [[holomorphic function]] on an [[open subset]] of the [[complex plane]] is called '''univalent''' if it is [[Injective function|injective]].

One can prove that if <math>G</math> and <math>\Omega</math> are two open [[connected space|connected]] sets in the complex plane, and

:<math>f: G \to \Omega</math>

is a univalent function such that <math>f(G) = \Omega</math> (that is, <math>f</math> is [[Surjective function|surjective]]), then the derivative of <math>f</math> is never zero, <math>f</math> is [[invertible]], and its inverse <math>f^{-1}</math> is also holomorphic. More, one has by the [[chain rule]]

Alternate terms in common use are ''schlicht''( this is German for plain, simple) and ''simple''. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.

==Important theorems==

===Riemann mapping theorem===

Let <math>z_0</math> be a point in a simply-connected region <math>D_1 (D_1 \neq \mathbb{C})</math> and <math>D_1</math> having at least two boundary points. Then there exists a unique analytic function <math>w=f(z)</math> mapping <math>D_1</math> bijectively into the open unit disk <math>|w| < 1</math> such that <math>f(z_0)=0</math> and <math>f'(z_0) > 0</math>.

Although [[Riemann's mapping theorem]] demonstrates the existence of a mapping function, it does not actually ''exhibit'' this function. An example is given below.

[[File:Illustration of Riemann Mapping Theorem.JPG|Illustration of Riemann Mapping Theorem]]

In the above figure, consider <math>D_1</math> and <math>D_2</math> as two simply connected regions different from <math>\mathbb C</math>. The [[Riemann mapping theorem]] provides the existence of <math>w=f(z)</math> mapping <math>D_1</math> onto the unit disk and existence of <math>w=g(z)</math> mapping <math>D_2</math> onto the unit disk. Thus <math>g^{-1}f</math> is a one-to-one mapping of <math>D_1</math> onto <math>D_2</math>.

===Elaboration===

If we can show that <math>g^{-1}</math>, and consequently the composition, is analytic, we then have a conformal mapping of <math>D_1</math> onto <math>D_2</math>, proving "any two simply connected regions different from the whole plane <math>\mathbb C</math> can be mapped conformally onto each other."

In the above figure, consider ''D''{{su|b=1}} and ''D''{{su|b=2}} as two simply connected regions different from ℂ. The [[Riemann mapping theorem]] provides the existence of ''w=f(z)'' mapping ''D''{{su|b=1}} onto the unit disk and existence of ''w=g(z)'' mapping ''D''{{su|b=2}} onto the unit disk. Thus ''g''{{su|p=-1}}''f'' is a one-one mapping of ''D''{{su|b=1}} onto ''D''{{su|b=2}}.

If we can show that ''g''{{su|p=-1}}, and consequently the composition, is analytic, we then have a conformal mapping of ''D''{{su|b=1}} onto ''D''{{su|b=2}}, proving "any two simply connected regions different from the whole plane ℂ can be mapped conformally onto each other."

==Univalent=Schwarz's functionLemma===

{{main|Schwarz lemma}}

Of special interest are those complex functions which are one-to-one. That is, for points ''z''{{su|b=1}}, ''z''{{su|b=2}}, in a ___domain ''D'', they share a common value, ''f(z{{su|b=1}})''=''f(z{{su|b=2}})'' only if they are the same point (''z''{{su|b=1}} = ''z''{{su|b=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''{{su|b=1}} and ''z''{{su|b=2}} in ''D'', i.e ''f(z{{su|b=1}})''≠''f(z{{su|b=2}})'' implies ''z''{{su|b=1}}≠''z''{{su|b=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.

The '''Schwarz lemma''', named after [[Hermann Amandus Schwarz]], is a result in [[complex analysis]] about [[holomorphic functions]] from the [[open set|open]] [[unit disk]] to itself. The lemma is less celebrated than stronger theorems, such as the [[Riemann mapping theorem]], which it helps to prove. It is however one of the simplest results capturing the rigidity of holomorphic functions.

====Statement====

<blockquote>'''Schwarz Lemma.''' Let '''D''' = {''z'' : |''z''| < 1} be the open [[unit disk]] in the [[complex number|complex plane]] '''C''' centered at the [[origin (mathematics)|origin]] and let ''f'' : '''D''' → '''D''' be a [[holomorphic map]] such that ''f''(0) = 0.

Then, |''f''(''z'')| ≤ |''z''| for all ''z'' in '''D''' and |''f′''(0)| ≤ 1.

Moreover, if |''f''(''z'')| = |''z''| for some non-zero ''z'' or if |''f′''(0)| = 1, then ''f''(''z'') = ''az'' for some ''a'' in '''C''' with |''a''| (necessarily) equal to 1.</blockquote>

===Maximum principle===

{{main|Maximum principle}}

The [[maximum principle]] is a property of solutions to certain [[partial differential equations]], of the [[elliptic partial differential equation|elliptic]] and [[parabolic partial differential equation|parabolic]] types. Roughly speaking, it says that the [[maximum]] of a function in a [[Domain (mathematical analysis)|___domain]] is to be found on the boundary of that ___domain. Specifically, the ''strong'' maximum principle says that if a function achieves its maximum in the interior of the ___domain, the function is uniformly a constant. The ''weak'' maximum principle says that the maximum of the function is to be found on the boundary, but may re-occur in the interior as well. Other, even weaker maximum principles exist which merely bound a function in terms of its maximum on the boundary.

===Riemann-Hurwitz formula===

{{main|Riemann-Hurwitz formula}}

the '''Riemann–Hurwitz formula''', named after [[Bernhard Riemann]] and [[Adolf Hurwitz]], describes the relationship of the [[Euler characteristic]]s of two [[Surface (topology)|surface]]s when one is a ''ramified covering'' of the other. It therefore connects [[Ramification (mathematics)|ramification]] with [[algebraic topology]], in this case. It is a prototype result for many others, and is often applied in the theory of [[Riemann surface]]s (which is its origin) and [[algebraic curve]]s.

==== Statement ====

For an [[orientable]] surface ''S'' the Euler characteristic χ(''S'') is

:<math>2-2g \,</math>

where ''g'' is the [[genus (mathematics)|genus]] (the ''number of handles''), since the [[Betti number]]s are 1, 2''g'', 1, 0, 0, ... . In the case of an (''unramified'') [[covering map]] of surfaces

:<math>\pi:S' \to S \,</math>

that is surjective and of degree ''N'', we should have the formula

:<math>\chi(S') = N\cdot\chi(S). \,</math>

That is because each simplex of ''S'' should be covered by exactly ''N'' in ''S''&prime; &mdash; at least if we use a fine enough [[Triangulation (geometry)|triangulation]] of ''S'', as we are entitled to do since the Euler characteristic is a [[topological invariant]]. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (''sheets coming together'').

Now assume that ''S'' and ''S&prime;'' are [[Riemann surface]]s, and that the map π is [[analytic function|complex analytic]]. The map π is said to be ''ramified'' at a point ''P'' in ''S''&prime; if there exist analytic coordinates near ''P'' and π(''P'') such that π takes the form π(''z'') = ''z''^''n'', and ''n''&nbsp;>&nbsp;1. An equivalent way of thinking about this is that there exists a small neighborhood ''U'' of ''P'' such that π(''P'') has exactly one preimage in ''U'', but the image of any other point in ''U'' has exactly ''n'' preimages in ''U''. The number ''n'' is called the ''[[ramification index]] at P'' and also denoted by ''e''_''P''. In calculating the Euler characteristic of ''S''&prime; we notice the loss of ''e_P''&nbsp;&minus;&nbsp;1 copies of ''P'' above π(''P'') (that is, in the inverse image of π(''P'')). Now let us choose triangulations of ''S'' and ''S&prime;'' with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then ''S&prime;'' will have the same number of ''d''-dimensional faces for ''d'' different from zero, but fewer than expected vertices. Therefore, we find a "corrected" formula

:<math>\chi(S') = N\cdot\chi(S) - \sum_{P\in S'} (e_P -1) </math>

(all but finitely many ''P'' have ''e_P'' = 1, so this is quite safe). This formula is known as the ''Riemann–Hurwitz formula'' and also as '''Hurwitz's theorem'''.

==References==

{{Reflist}}

* {{Citation |last=Ahlfors |first=Lars |author-link=Lars Ahlfors |title=Zur Theorie der Überlagerungsflächen |journal=[[Acta Mathematica]] |volume=65 |issue=1 |pages=157–194 |year=1935 |issn=0001-5962 |language=de |doi=10.1007/BF02420945 |jfm=61.0365.03 |zbl=0012.17204 |doi-access=free}}.

*{{Citation |last=Grötzsch |first=Herbert |author-link=Herbert Grötzsch |title=Über einige Extremalprobleme der konformen Abbildung. I, II. |language=de |year=1928 |journal=[[Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Classe]] |volume=80 |pages=367–376, 497–502 |jfm=54.0378.01}}.

* Hurwitz-Courant, ''Vorlesunger über allgemeine Funcktionen Theorie'', 1922 (4th ed., appendix by H. Röhrl, vol. 3, ''Grundlehren der mathematischen Wissenschaften''. Springer, 1964.)

*{{cite book |title=Geometric Function Theory: Explorations in Complex Analysis|

first=Steven|last=Krantz|publisher=Springer|year=2006|isbn=0-8176-4339-7}}

*{{Cite journal | last1 = Bulboacă | first1 = T. | last2 = Cho | first2 = N. E. | last3 = Kanas | first3 = S. A. R. | title = New Trends in Geometric Function Theory 2011 | doi = 10.1155/2012/976374 | journal = International Journal of Mathematics and Mathematical Sciences | volume = 2012 | pages = 1–2 | year = 2012 | url = http://downloads.hindawi.com/journals/ijmms/2010/906317.pdf | doi-access = free }}

*{{cite book |title=Lecture notes on Introduction to Univalent Functions|

*{{cite book | isbn = 978-0821852705 | title = Conformal Invariants: Topics in Geometric Function Theory | last1 = Ahlfors | first1 = Lars | year = 2010 | publisher = AMS Chelsea Publishing }}

first=K.I|last=Noor|publisher=CIIT, Islamabad, Pakistan}}

{{mathanalysis-stub}}

[[Category:Analytic functions|*]]