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4434Escher and the Droste Effect
http://www.merlot.org/merlot/viewMaterial.htm?id=83200
The goal of this site is to visualize the mathematical structure behind M.C. Escher's picture called "Print Gallery" (1956). The visualization itself is largely non-mathematical and is accomplished through many still images and animations. The actual mathematics, involving conformal mappings of the complex plane, is contained in a pdf copy of the original AMS publication. The Droste Effect refers to any image that contains itself on a smaller scale.Graphics for Complex Analysis
http://www.merlot.org/merlot/viewMaterial.htm?id=89786
A large set of animations designed to display fundamental and specialized topics in complex analysis.Walter Fendt's Mathematics Applets
http://www.merlot.org/merlot/viewMaterial.htm?id=91208
A large collection of applets designed to demonstrate basic mathematics. Many foreign-language versions available.Stereographic Projection
http://www.merlot.org/merlot/viewMaterial.htm?id=374828
The site contains a JAVA applet that illustrates the Stereographic Projection map of the sphere to the plane. The applet allows the pattern on the sphere to be rotated so that the stretching of its projection on the plane can be examined. This site is a part of a large collection of materials related to geometry and scientific graphing.Dave's Short Course on Complex Numbers
http://www.merlot.org/merlot/viewMaterial.htm?id=83731
This is a nicely rendered introduction to complex numbers at the high school or freshman college level. Reference is made to another of Dave's short courses in trigonometry.A Complex Function Viewer
http://www.merlot.org/merlot/viewMaterial.htm?id=82940
This Java applet allows the user to visualize certain functions of complex variable. The functions considered are complex sine, cosine, and exponential function.Mathematics Animated
http://www.merlot.org/merlot/viewMaterial.htm?id=84907
This site contains more than three dozen animations on various topics including the Pythagorean Theorem, the unit circle as used in defining trig functions, inversion and squaring in the complex plane, and many others pertaining to calculus and its associated analytic geometry.A First Course in Complex Analysis
http://www.merlot.org/merlot/viewMaterial.htm?id=518850
According to OER Commons, 'These are the lecture notes of a one-semester undergraduate course which we taught at SUNY Binghamton. For many of our students, Complex Analysis is their first rigorous analysis (if not mathematics) class they take, and these notes reflect this very much. We tried to rely on as few concepts from real analysis as possible. In particular, series and sequences are treated "from scratch." This also has the (maybe disadvantageous) consequence that power series are introduced very late in the course.'Complex Analysis
http://www.merlot.org/merlot/viewMaterial.htm?id=437630
This is a free, online textbook for an introductory course in complex analysis. General topics include Complex Numbers, Complex Functions, Elementary Functions, Integration, Cauchy's Theorem, More Integration, Harmonic Functions, Series, Taylor and Laurent Series, Poles, Residues, and All That, and Argument Principle. Each chapter from the book can be downloaded as a free pdf file.Complex Analysis
http://www.merlot.org/merlot/viewMaterial.htm?id=731702
This is a free online course offered by the Saylor Foundation.'This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable. Such functions, beautiful on their own, are immediately useful in Physics, Engineering, and Signal Processing. Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Simply put, you will be working with lines and sets and very specific functions on the complex plane—drawing pictures of them and teasing out all of their idiosyncrasies. You will again find yourself calculating line integrals, just as in multivariable calculus. However, the techniques you learn in this course will help you get past many of the seeming dead-ends you ran up against in calculus. Indeed, most of the definite integrals you will learn to evaluate in Unit 7 come directly from problems in physics and cannot be solved except through techniques from complex variables.We will begin by studying the minimal algebraically closed extension of real numbers: the complex numbers. The Fundamental Theorem of Algebra states that any non-constant polynomial with complex coefficients has a zero in the complex numbers. This makes life in the complex plane very interesting. We will also review a bit of the geometry of the complex plane and relevant topological concepts, such as connectedness.In Unit 2, we will study differential calculus in the complex domain. The concept of analytic or holomorphic function will be introduced as complex differentiability in an open subset of the complex numbers. The Cauchy-Riemann equations will establish a connection between analytic functions and differentiable functions depending on two real variables. In Unit 3, we will review power series, which will be the link between holomorphic and analytic functions. In Unit 4, we will introduce certain special functions, including exponentials and trigonometric and logarithmic functions. We will consider the Möbius Transformation in some detail.In Units 5, 6, and 7 we will study Cauchy Theory, as well as its most important applications, including the Residue Theorem. We will compute Laurent series, and we will use the Residue Theorem to evaluate certain integrals on the real line which cannot be dealt with through methods from real variables alone. Our final unit, Unit 8, will discuss harmonic functions of two real variables, which are functions with continuous second partial derivatives that satisfy the Laplace equation, conformal mappings, and the Open Mapping Theorem.'