Graduate Mathematical Physics, With MATHEMATICA Supplements - Softcover

Kelly, James J.

Softcover

ISBN 10: 3527406379 ISBN 13: 9783527406371

Publisher: Wiley-VCH, 2006

Synopsis

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/

About this title

About the Author

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.

Excerpt. © Reprinted by permission. All rights reserved.

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] (1.19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)

In the Argand plane, these roots are found at the vertices of a regular n-sided polygon inscribed within the unit circle with the principal root at z = 1. More generally, the roots

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)

of [e.sup.i]theta]] are found at the vertices of a rotated polygon inscribed within the unit circle, as illustrated in Fig. 1.4.

1.1.4 Argument Function

The graphical representation of complex numbers suggests that we should obtain the phase using

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)

but this definition is unsatisfactory because the ratio y/x is not sensitive to the quadrant, being positive in both first and third and negative in both second and fourth quadrants. Consequently, computer programs using arctan [y/x] return values limited to the range (-[pi]/2, [pi]/2).

A better definition is provided by a quadrant-sensitive extension of the usual arctangent function

ArcTan [x,y] = ArcTan[y/x] + [pi]/2 (1-Sign [x])Sign[y] (1.24)

that returns values in the range (-[pi], [pi]). (Unfortunately, the order of the arguments is reversed between Fortran and Mathematica.) Therefore, we define the principal branch of the argument function by

Arg[z] = Arg[x + iy] = ArcTan [x, y] (1.25)

where -[pi] < Arg[z] [less than or equal to] [pi].

However, the polar representation of complex numbers is not unique because the phase [theta] is only defined modulo 2[pi]. Thus,

arg[z] = Arg[z] + 2[pi]n (1.26)

is a multivalued function where n is an arbitrary integer. Note that some authors distinguish between these functions by using lower case for the multivalued and upper case for the single-valued version while others rely on context. Consider two points on opposite sides of the negative real axis, with y [right arrow] [0.sup.+] infinitesimally above and y [right arrow] [0.sup.-] infinitesimally below. Although these points are very close together, Arg[z] changes by 2[pi] across the negative real axis.

A discontinuity of this type is usually represented by a branch cut. Imagine that the complex plane is a sheet of paper upon which axes are drawn. Starting at the point (r, 0) one can reach any point z = (x, y) by drawing a continuous circular arc of radius r = [square root of [x.sup.2] + [y.sup.2] and we define arg[z] as the angle subtended by that arc. This function is multivalued because the circular arc can be traversed in either direction or can wind around the origin an arbitrary number of times before stopping at its destination. A single-valued version can be created by making a cut infinitesimally below the negative real axis, as sketched in Fig. 1.5, that prevents a continuous arc from subtending more than [pi] radians. Points on the negative real axis are reached by positive (counterclockwise) arcs with Arg[(-[absolute value of z], 0)] = [pi] while points infinitesimally below the negative real axis can only be reached by negative arcs with Arg[(-[absolute value of z], [0.sup.-])] [right arrow] [right arrow] -[pi]. Thus, Arg[z] is single-valued and is continuous on any path that does not cross its branch cut, but is discontinuous across the cut.

The principal branch of the argument function is defined by the restriction -[pi] < Arg[z] [less than or equal to] [pi]. Notice that one side of this range is open, represented by <, while the other side is closed, represented by [less than or equal to]. This notation indicates that the cut is infinitesimally below the negative real axis, such that the argument for negative real numbers is [pi], not -[pi]. This choice is not unique, but is the nearly universal convention for the argument and many related functions. The distinction between < and [less than or equal to] many seem to be nitpicking, but attention to such details is often important in performing accurate derivations and calculations with functions of complex variables.

Many functions require one or more branch cuts to establish single-valued definitions; in fact, handling either the multivaluedness of functions of complex variables or the discontinuities associated with their single-valued manifestations is often the most difficult problem encountered in complex analysis. Although our choice of branch cut for Arg[z] is not unique (any radial cut from the origin to [infinity] would serve the same purpose), it is consistent with the customary definitions of ArcTan, Log, and other elementary functions to be discussed in more detail later. The single-valued version of a function that is most common is described as its principal branch. For many functions there is considerable flexibility in the choice of branch cut and we are free to make the most convenient choice, provided that we maintain that choice throughout the problem. For example, in some applications it might prove convenient to define an argument function with the range -3[pi]/4 < MyArg[z] [less than or equal to] 5[pi]/4 using the branch cut shown in Fig. 1.6. Consider the point [z.sub.1] (-1, -1) for which the standard argument function gives Arg[z.sub.1] = 3[pi]/4 while our new argument function gives MyArg[z.sub.1] = 5[pi]/4. These functions are obviously different because the same input gives different output, but both represent precisely the same ray in the complex plane. Therefore, we should consider the specification of the branch cuts as an important part of the definition of a single-valued function and recognize that different choices of cuts lead to related but different functions.

It is important to recognize that, because of discontinuities across branch cuts, simple algebraic relationships that apply to multivalent functions of complex variables, often do not pertain to their monovalent cousins. For example, using the polar representation of the product of two complex numbers we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.27)

but this relationship for the phase does not necessarily apply to the principal branch because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.28)

where n must be chosen to ensure that -[pi] < Arg[[z.sub.1][z.sub.2]] [less than or equal to] [pi]. Often the price of single-valuedness is the awkwardness of discontinuities.

1.2 Take Care with Multivalued Functions

Ambiguities in the definitions of many seemingly innocuous functions require considerable care. For example, consider the common replacement

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.29)

that one often makes without thinking. Is this apparent equivalence correct? Compare the following two methods for evaluating these quantities when z [right arrow] -1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.31)

Both calculations look correct, but their results differ in sign. These expressions are not always interchangeable! One must take more care with multivalued functions.

If we represent the complex number z in Cartesian form z = x + iy where x, y are real, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.32) (Continues...)

Excerpted from Graduate Mathematical Physics, With MATHEMATICA Supplementsby James J. Kelly Copyright © 2006 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Excerpted by permission.
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Edition details

• Publisher: Wiley-VCH
• Publication date: 2006
• Language: English
• ISBN 10: 3527406379
• ISBN 13: 9783527406371
• Binding: Paperback
• Edition number: 1
• Number of pages: 482