Examples of diff/cont at a point

Cont only at a point

f(z) = |z| if |z| irrational and -|z| if |z| is irrational. (Rotate usual function.)
Note that if z_0 =/= 0 then lim_{z->z_0}f(z) DNE. Given \epsilon > 0 consider |f(z)-f(z_0)| for z\in B(\epsilon,z_0). If z_0 irrat then z->z_0 along rational approximations gives |f(z)-f(z_0)|=2|z_0|+\epsilon -/->0 as z->z_0. So f is not cts at z_0.

As z->0 |f(z)-f(0)|=|z_0|->0. So cts at 0.

Diff only at a pt. Example in lectures: |x| I think.

Cts *and* diffable only at a pt? f(z)=|z|^1/2 (|z| rational), -|z|^1/2 (|z| irrational). Cts at zero. Not cts away from zero. Let z_0 have |z_0| rational.

|f(z)-f(z_0)/z-z_0|=| +/- |z_0+\epsilon|^1/2- |z_0|^1/2 | / |epsilon|

= (same top line)/(|(z_0+\epsilon)|^1/2 - |z_0|^1/2)(|(z_0+\epsilon)|^1/2 + |z_0|^1/2

Nope, this is not the case! more work necessary...

lim_z_0->0 |\epsilon|^1/2 / |epsilon| which ->0 as \epsilon -> 0.

A Brief Tutorial

A Brief Tutorial

Overview

Simply stated, Bombelli is just another computer programme for plotting complex functions. Nevertheless, it features many characteristics that are at least unusual amongst other applications at its level, namely:

* It is gratuitous;
* On account of having been written in JAVA, it is platform independent and avaiable through the Internet;
* It offers a high degree of flexibility on defining new functions such as polymomial, trigonometric, exponential, Möbius Transformations, etc.;
* It includes four types of domain shapes: square grid, circular grid, Arnold's cat and a free-form coninuous curve;
* It permits fine detail observation through the "scroll" and "zoom" controls;
* It offers an adjustable degree of "smoothness".

A Complex Function Viewer (Java)

A Complex Function Viewer (Java): "This Java applet allows you to visualize certain maps from the Complex plane to itself. Eventually, we'd like to be able to visualize more general maps from the Cartesian plane to itself. We welcome feedback on this project.

Use the mouse pointer to move the small (red) square grid around. It represents a neighbourhood of a point in the domain. The twisted (blue) grid represents the image of the square grid."

To investigate soon...

Chinn and Steenrod's popular-ish book, including discussion of winding number etc.

Chinn, William G.
Title: First concepts of topology : the geometry of mappings of segments, curves, circles, and disks / by W. G. Chinn and N. E. Steenrod
Published: [New York] : Random House, [1966]
Description: viii, 160 p. : illus ; 23 cm.
Other Author(s): Steenrod, Norman Earl, 1910-1971, joint author
Subject Heading(s): Topology.
Series Title(s): New mathematical library, 18

Location: Barr Smith Main collection
Call Number: 515.1 C539f
Status: Not on loan

Complex Analysis

A free textbook on Complex Analysis (useable??)


Complex Analysis: "Complex Analysis
by
George Cain
(c)Copyright 1999, 2001 by George Cain. All rights reserved.
This is a textbook for an introductory course in complex analysis. It has been used for our undergraduate complex analysis course here at Georgia Tech and at a few other places that I know of.

I owe a special debt of gratitude to Professor Matthias Beck of SUNY Binghamton who used the book in his class at Binghamton and found many errors and made many good suggestions for changes and additions to the book. I thank him very much. I have corrected the errors and made some changes.

I am also grateful to Professor Pawel Hitczenko of Drexel University, who prepared the nice supplement to Chapter 10 on applications of the Residue Theorem to real integration.

The notes are available as Adobe Acrobat documents. If you do not have an Adobe Acrobat Reader, you may down-load a copy, free of charge, from Adobe."

Notes for Conformal Maps

Maple has a nice series of packages for viewing conformal maps: in particular, the "Maple Book" on p282 lists the Joukowski plot for airfoils. First they define a proc for JoukowskiP and then use it like...

with(plots);
display(JoukowskiP(1/10,1/10),scaling=constrained,thickness=2);