Graduate Mathematical Physics: With MATHEMATICA Supplements - Softcover

Kelly, James J.

Softcover

ISBN 10: 3527406379 ISBN 13: 9783527406371

Verlag: Wiley-VCH, 2006

Inhaltsangabe

This up-to-date textbook on mathematical methods of physics is designed for a one-semester graduate or two-semester advanced undergraduate course. The formal methods are supplemented by applications that use MATHEMATICA to perform both symbolic and numerical calculations.
The book is written by a physicist lecturer who knows the difficulties involved in applying mathematics to real problems. As many as 40 exercises are included at the end of each chapter. A student CD includes a basic introduction to MATHEMATICA, notebook files for each chapter, and solutions to selected exercises.
 
* Free solutions manual available for lecturers at www.wiley-vch.de/supplements/

Über diesen Titel

Über die Autorin bzw. den Autor

James Kelly received his BS from CalTech in 1977 and his PhD from MIT in 1981. He was an Oppenheimer Fellow at Los Alamos before joining the faculty of the University of Maryland in 1984, where he is currently a Professor. His research is primarily in experimental nuclear physics, where he is expert in data analysis and simulation, but he also often performs the calculations required to test theoretical models. His most recent topic is the electromagnetic structure of the nucleon and its low-lying excited states.

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Graduate Mathematical Physics, With MATHEMATICA Supplements

By James J. Kelly

John Wiley & Sons

Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
All right reserved.
ISBN: 978-3-527-40637-1

Chapter One

Analytic Functions

Abstract. We introduce the theory of functions of a complex variable. Many familiar functions of real variables become multivalued when extended to complex variables, requiring branch cuts to establish single-valued definitions. The requirements for differentiability are developed and the properties of analytic functions are explored in some detail. The Cauchy integral formula facilitates development of power series and provides powerful new methods of integration.

1.1 Complex Numbers

1.1.1 Motivation and Definitions

The definition of complex numbers can be motivated by the need to find solutions to polynomial equations. The simplest example of a polynomial equation without solutions among the real numbers is [z.sup.2] = -1. Gauss demonstrated that by defining two solutions according to

[z.sup.2] = -1 [??]z = i (1.1)

one can prove that any polynomial equation of degree n has n solutions among complex numbers of the form z = x + iy where x and y are real and where [i.sup.2] = -1. This powerful result is now known as the fundamental theorem of algebra. The object i is described as an imaginary number because it is not a real number, just as [square root of 2] is an irrational number because it is not a rational number. A number that may have both real and imaginary components, even if either vanishes, is described as complex because it has two parts. Throughout this course we will discover that the rich properties of functions of complex variables provide an amazing arsenal of weapons to attack problems in mathematical physics.

The complex numbers can be represented as ordered pairs of real numbers z = (x, y) that strongly resemble the Cartesian coordinates of a point in the plane. Thus, if we treat the numbers 1 = (1, 0) and i = (0, 1) as basis vectors, the complex numbers z = (x, y) = x x 1 + y x + iy can be represented as points in the complex plane, as indicated in Fig. 1.1. A diagram of this type is often called an Argand diagram. It is useful to define functions Re or Im that retrieve the real part x = Re [z] or the imaginary part y = Imz of a complex number. Similarly, the modulus, r, and phase, [theta], can be defined as the polar coordinates

r = [square root of [x.sup.2] + [y.sup.2]], [theta] = ArcTan [y/x] (1.2)

by analogy with two-dimensional vectors.

Continuing this analogy, we also define the addition of complex numbers by adding their components, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

as diagrammed in Fig. 1.2. The complex numbers then form a linear vector space and addition of complex numbers can be performed graphically in exactly the same manner as for vectors in a plane.

However, the analogy with Cartesian coordinates is not complete and does not extend to multiplication. The multiplication of two complex numbers is based upon the distributive property of multiplication

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

and the definition [i.sup.2] = -1. The product of two complex numbers is then another complex number with the components

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

More formally, the complex numbers can be represented as ordered pairs of real numbers z = (x, y) with equality, addition, and multiplication defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

One can show that these definitions fulfill all the formal requirements of a field, and we denote the complex number field as C. Thus, the field of real numbers is contained as a subset, R [subset] C.

It will also be useful to define complex conjugation

complex conjugation: z = (x, y) [??] [z.sup.*] = (x, - y) (1.9)

and absolute value functions

absolute value: [absolute value of z] = [square root of [x.sup.2] + [y.sup.2]] (1.10)

with conventional notations. Geometrically, complex conjugation represents reflection across the real axis, as sketched in Fig. 1.3.

The Re, Im, and Abs functions can now be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

Thus, we quickly obtain the following arithmetic facts:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

1.1.2 Triangle Inequalities

Distances between points in the complex plane are calculated using a metric function. A metric d [a, b] is a real-valued function such that

1. d[a, b] > 0 for all a [not equal to] b

2. d[a, b] = 0 for all a = b

3. d[a, b] = d[a, b]

4. d[a, b] [less than or equal to] d[a, c] + d[c, b] for any c.

Thus, the Euclidean metric d][z.sub.1], [z.sub.2] = [absolute value of [[z.sub.1] - [z.sub.2]] = [square root of [([x.sub.1] - [x.sub.2]).sup.2] + [([y.sub.1] - [y.sub.2].sup.2] is suitable for [??]. Then with geometric reasoning one easily obtains the triangle inequalities:

triangle inequalities: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

Note that C cannot be ordered (it is not possible to define < properly).

1.1.3 Polar Representation

The function [e.sup.i]theta]] can be evaluated using the power series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

giving a result known as Euler's formula. Thus, we can represent complex numbers in polar form according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

where r is the modulus or magnitude and [theta] is the phase or argument of z. Although addition of complex numbers is easier with the Cartesian representation, multiplication is usually easier using polar notation where the product of two complex numbers becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.17)

Thus, the moduli multiply while the phases add. Note that in this derivation we did not assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which we have not yet proven for complex arguments, relying instead upon the Euler formula and established properties for trigonometric functions of real variables.

Using the polar representation, it also becomes trivial to prove de Moivre's theorem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18)

However, one must be careful in performing calculations of this type. For example, one cannot simply replace [([e.sup.in[theta]]).sup.1/n] by [e.sup.i]theta]] because the equation, [z.sup.n] = w has n solutions {[z.sub.k], k = 1, n} while [e.sup.i[theta]] is a unique complex number. Thus, there are n, nth-roots of unity, obtained as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]...

Details zur Ausgabe

• Verlag: Wiley-VCH
• Erscheinungsdatum: 2006
• Sprache: Englisch
• ISBN 10: 3527406379
• ISBN 13: 9783527406371
• Einband: Taschenbuch
• Auflage: 1
• Anzahl der Seiten: 482
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Buchpark GmbH; gpsr@buchpark.de; ; Krügerweg 1; Trebbin; Brandenburg; 14959; Deutschland