Complex analysis ii spring 2015 these are notes for the graduate course math 5293 complex analysis ii taught by dr. Cas representing a point or a vector x,y in r2, and according to. We will extend the notions of derivatives and integrals, familiar from calculus. Rudolf wegmann, in handbook of complex analysis, 2005. A universal cover of any suitable space x is a simply connected space which maps to x via a covering map.
This document includes complete solutions to both exams in 20, as well as select solutions from some older exams. C is called simply connected if the complement in the extended complex plane is also connected. Simply connected region an overview sciencedirect topics. Assuming only undergraduate real analysis and following the power series approach, it quickly and elegantly develops the basic theory through cauchys theorem for cycles, normal families, the riemann mapping theorem, and the weierstrass and mittagleffler theorems. Openness by the above argument just center disc around any. Sometimes we will call c the open complex plane in order to stress the di. Anthony kable at the oklahoma state university spring 2015. Cauchys integral formula states that every function holomorphic inside a disk is completely determined by its values on the disks boundary. Preliminaries to complex analysis the complex numbers is a eld c. Simplyconnected, complex, simple lie groups physics forums. This implies the general harmonic functions have the mean value property too, by linearity. Calculus iii fundamental theorem for line integrals. A region \d\ is connected if we can connect any two points in the region with a path that lies completely in \d\.
If x and y are homotopy equivalent and x is simply connected, then so is y. Some solutions to the exercises in ssh03 are also written down. As sets of the type described above play an important role in complex analysis, we shall nd it convenient to give them a name. For twodimensional regions, a simply connected domain is one without holes in it. The problems are organized in reverse chronological order, so the most recent exams appear rst. Complex analysis ii oklahoma state universitystillwater. Chapter 2 complex analysis in this part of the course we will study some basic complex analysis. A region \d\ is simplyconnected if it is connected and it contains no holes.
This is an even stronger condition that pathconnected. Let u c be a nonempty, open, simply connected set, and f. There are nsolutions as there should be since we are. We will see that this is a simple consequence of the cauchyriemann equations. For the love of physics walter lewin may 16, 2011 duration. In the work on greens theorem so far, it has been assumed that the region r has as its boundary a single simple closed curve. Aregion d is said to be simply connected if any simple closed curve which lies entirely in d can be pulled to a single point in d a curve is called simple if it has no self intersections. Solutionshints to the exercises from complex analysis by stein and shakarchi 3 solution 3. In the textbook of complex analysis i have, the author defined the definition of simply connected region as follows. The image of a simply connected set under a continuous function need not be simply connected.
Definition a set is simplyconnected if any loop completely contained in can be shrunk down to a point without leaving. Complex analysis connected sets and curves youtube. Complex analysis lecture notes uc davis mathematics. Another related notion is locally connected, which neither implies nor follows from connectedness. These notes are about complex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. Some related but stronger conditions are path connected, simply connected, and n connected. In fact, to a large extent complex analysis is the study of analytic functions. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. There are at least two reasons why harmonic functions are part of the study of complex analysis, and they can be found in the next two theorems.
Pdf on the structure of complete simplyconnected kahler. In the first chapter, he says dynkin diagrams classify various types of object, including simplyconnected, complex, simple lie groups. We wont need this one until the next section, but it fits in with all the other definitions given here so this was a natural place to put the. I have some question on the definition of simply connected region in complex analysis. A connected open subset u of the plane r2 is said to be simply connected in the. An alternative slicker proof is to use the definition of connected to show that a connected set has no proper subset that is both relatively open and relatively closed within the connected set. Solutions to complex analysis prelims ben strasser in preparation for the complex analysis prelim, i typed up solutions to some old exams. A simply connected domain is a path connected domain where one can continuously shrink any simple closed curve into a point while remaining in the domain. Pdf study of simply connected domain and its geometric. A nonempty open, connected hence polygonally connected subset of the complex plane is called a region. Analysis, real and complex analysis, and functional analysis, whose widespread use is illustrated by the fact that they have been translated into a total of languages. Rational numbers can be thought of as points on an in nite horizontal line once one has speci ed the positions of 0 and 1 with 1 lying to the right hand side of the point 0.
Complex analysis, one of the genuine masterpieces of the subject. Two definition of simply connected region in complex analysis. We talked about the properties of complex numbers, how we add and multiply them, their conjugate, their polar form and how the polar form interacts with multiplication. While this may sound a bit specialized, there are at least two excellent reasons why all mathematicians should learn about complex analysis. Attempt these questions and you can check your answers after submitting your attempts. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. By applying the above remarks to f we know that f and hence uhave the mean value property on dp. The set is not connected, either, since if you take a point in the inner disk, and another point in the outer ring, there is no path between them which does not leave the set. Complex analysis mcqs test 02 consist of multiple choice questions. Note however, that if the path connectedness requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected.
On the other hand, the complement of this set in rn is home. Download an introduction to complex analysis by ravi p. Dec 29, 2016 ive been looking at john baezs lecture notes lie theory through examples. Since a simply connected space is, by definition, also required to be path connected, any simply connected space is also connected. Basye the class of simply connected sets, which is the object of study of the present paper, is closely related to the class of unicoherent sets introduced by vietorisj and kuratowski. For a region to be simply connected, in the very least it must be a region i. Suppose the region has a boundary composed of several simple closed curves, like the. Sandra pinelas this textbook introduces the subject of complex analysis to advanced undergraduate and graduate students in a clear and concise manner.
Notes on connected and disconnected sets in this worksheet, well learn about another way to think about continuity. Each simply connected region g in the extended complex plane c. Jan 19, 2012 for the love of physics walter lewin may 16, 2011 duration. Show that the complex plane minus a finite nonempty set is not simply connected. A set is said to be simply connected if every closed path in the set is homotopic to a point. The set of complex numbers with imaginary part strictly greater than zero and less than one, furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected. I do not claim that the notes or solutions written here are correct or elegant.
However, one can show by using techniques outside of complex analysis just mean value property of solutions to laplaces equation that. Riemann formulated in his famous thesis 235 a remarkable mapping theorem which in modem language reads. Lecture11 characterisations of simply connected sets recall that. We will say that an open subset u of the complex plane is connected, if every pair of. A subset of a topological space x is a connected set if it is a connected space when viewed as a subspace of x. A domain din c is simply connected if the inside of every simple closed curve in the domain is in the domain. Primarily, a familiarity with the concept of a connected set is of use. Cause if it says there must exist a point then the third set must be simply.