Chapter One
Complex Numbers
1.1 Introduction. Let us hark back to the first grade when the only numbers you knew were the ordinary everyday integers. You had no trouble solving problems in which you were, for instance, asked to find a number x such that 3
Chapter Seven
Harmonic Functions
7.1. The Laplace equation. The Fourier law of heat conduction says that the rate at which heat passes across a surface S is proportional to the flux, or surface integral, of the temperature gradient on the surface: k T dA.
Chapter Five
Cauchys Theorem
5.1. Homotopy. Suppose D is a connected subset of the plane such that every point of D is an interior pointwe call such a set a regionand let C 1 and C 2 be oriented closed curves in D. We say C 1 is homotopic to C 2 in D if t
Chapter Four
Integration
4.1. Introduction. If : D C is simply a function on a real interval D , , then the integral tdt is, of course, simply an ordered pair of everyday 3 rd grade calculus integrals:
tdt xtdt i ytdt,
where t xt iyt. Thus, for examp
Chapter Two
Complex Functions
2.1. Functions of a real variable. A function : I C from a set I of reals into the complex numbers C is actually a familiar concept from elementary calculus. It is simply a function from a subset of the reals into the plane,
Chapter Ten
Poles, Residues, and All That
10.1. Residues. A point z 0 is a singular point of a function f if f not analytic at z 0 , but is analytic at some point of each neighborhood of z 0 . A singular point z 0 of f is said to be isolated if there is a
Chapter Three
Elementary Functions
3.1. Introduction. Complex functions are, of course, quite easy to come bythey are simply ordered pairs of real-valued functions of two variables. We have, however, already seen enough to realize that it is those complex
Chapter Eleven
Argument Principle
11.1. Argument principle. Let C be a simple closed curve, and suppose f is analytic on C. Suppose moreover that the only singularities of f inside C are poles. If fz 0 for all zC, then fC is a closed curve which does not
Chapter Six
More Integration
6.1. Cauchys Integral Formula. Suppose f is analytic in a region containing a simple closed contour C with the usual positive orientation and its inside , and suppose z 0 is inside C. Then it turns out that f z 0 1 2i
C
fz z z
Chapter Nine
Taylor and Laurent Series
9.1. Taylor series. Suppose f is analytic on the open disk |z z 0 | r. Let z be any point in this disk and choose C to be the positively oriented circle of radius , where |z z 0 | r. Then for sC we have 1 1 1 s z s z
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Chapter Eight
f : Rn R
8.1 Introduction We shall now turn our attention to the very important special case of functions that are real, or scalar, valued. These are sometimes called scalar fields. In the very, but important, special subcase in which the di
Chapter Nine
The Taylor Polynomial
9.1 Introduction Let f be a function and let F be a collection of "nice" functions. The approximation problem is simply to find a function g F that is "close" to the given function f . There are two issues immediately. H
Chapter Ten
Sequences, Series, and All That
10.1 Introduction Suppose we want to compute an approximation of the number e by using the Taylor polynomial pn for f (x ) = e x at a =0. This polynomial is easily seen to be
p n ( x) = 1 + x + x2 x3 xn + +K+ .
Chapter Two
Vectors-Algebra and Geometry
2.1 Vectors A directed line segment in space is a line segment together with a direction. Thus the directed line segment from the point P to the point Q is different from the directed line segment from Q to P. We f
Chapter Six
Linear Functions and Matrices
6.1 Matrices Suppose f : R n R p be a linear function. Let e1 , e 2 ,K , e n be the coordinate vectors for R n . For any x R n , we have x = x 1e 1 + x 2 e 2 +K+ x n e n . Thus f ( x) = f ( x1 e 1 + x 2 e2 +K+ x n
Chapter Three
Vector Functions
3.1 Relations and Functions We begin with a review of the idea of a function. Suppose A and B are sets. The Cartesian product A B of these sets is the collection of all ordered pairs (a ,b) such that a A and b B . A relation
Chapter One
Euclidean Three-Space
1.1 Introduction. Let us briefly review the way in which we established a correspondence between the real numbers and the points on a line, and between ordered pairs of real numbers and the points in a plane. First, the l
Chapter Seven
Continuity, Derivatives, and All That
7.1 Limits and Continuity Let x 0 R n and r > 0 . The set B (a ; r ) = cfw_x R n :| x a| < r is called the open ball of radius r centered at x 0 . The closed ball of radius r centered at x 0 i s the set
Chapter Five
More Dimensions
5.1 The Space R n We are now prepared to move on to spaces of dimension greater than three. These spaces are a straightforward generalization of our Euclidean space of three dimensions. Let n be a positive integer. The n-dimen
Chapter Four
Derivatives
4.1 Derivatives Suppose f is a vector function and t 0 is a point in the interior of the domain of f ( t 0 in the interior of a set S of real numbers means there is an interval centered at t 0 that is a subset of S.). The derivati
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Since the major part of a small business typically comes from business to business services, it is essential to maintain a positive standing with the local business community