Mathematical methods for mathematicians, physical scientists, and engineers

124 Pages·5.709 MB·English

by  J. Dunning-Davies.

1
Introduction
1.1 COMPLEX NUMBERS
Roots of equations that are neither completely real nor completely imaginary are
often termed complex. By using the word imaginary, reference is made to roots of
negative numbers with V-l being the most popular example.
1.1.1 Early history
Complex numbers have a history that can be traced to work by Greek
mathematician, Heron of Alexandria, who lived sometime between 100 BC and
100 AD. They first appeared in a study concerned with the dimensions of a
pyramidal frustum. Although Heron of Alexandria recognised the conceptual
possibility of negative numbers possessing square roots, it took a considerable
period of time before they started to become of practical significance. This was owed
to discoveries made by Scipione del Ferro and Girolamo Cardano roughly between
1450 and 1600 AD. From 100 AD to the fifteenth century, very little information on
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imaginary numbers was recorded. Worthy of note are contributions made by
scholars such as Diophanrus of Alexandria (circa 300 AD) and Mahaviracarya (circa
850 AD) who both also considered the conceptual possibility of square roots of
negative numbers. By the eighteenth century complex numbers had achieved
considerable recognition and were starting to become written as, for example, 3 + 5i,
where 3 represents what is known as the real component and 5i is the imaginary
component. The letter / is representative of V-l and was first used by Euler in 1777.
To this day they have been expressed in this manner.
A considerable portion of this book relies on a geometric consideration of complex
numbers as opposed to an algebraic one. The geometry can be understood by
consideration of work by Wallis, Wessel and Argand which spans from the
seventeenth to the nineteenth century. This work was responsible for what are now
2 Introduction [Ch. 1
commonly known as Argand diagrams that represent complex numbers by an
imaginary y-axis and a real x-axis as illustrated in Fig. 2.2.1 for example. The first
notion that the y-axis should be positioned vertically with respect to the real axis was
provided by Wallis (1616-1703). For further information on the history of complex
numbers Nahin (1998) provides a clear account in a recent publication.
1.1.2 Complex function theory
Important ideas associated with this aspect of complex number theory were first
conceived by Augustin-Louis Cauchy (1789-1857) in a paper written in 1814 that
describes the integration of complex functions. Gauss was involved with similar
work at the same time that Cauchy published these findings. Riemann made use of
Cauchy's work in his doctoral dissertation in 1851. In Sections 3.3 and 3.4 complex
function integration is studied in some detail with reference to the xi function
defined by Riemann.
1.1.3 Practical applications
Complex numbers find widespread use in many scientific subject areas. They can be
applied in many branches of physics and also in astronomy to monitor planetary
motion. Electrical engineers use complex numbers to assess and evaluate electronic
circuitry and any science student is likely to first encounter applications of complex
numbers in this context as well. Within electrochemistry, which is a branch of
chemistry often concerned with characterisation and optimisation of electrical
devices such as batteries, fuel cells and sensors, the complex algebra linked to AC
impedance theory can be used to rationalise ways in which devices perform as they
operate (Roy, 1996). For example, the manner in which a battery generates power
can be modelled by a combination of resistors and capacitors. The AC impedance of
this model replicates the battery performance and therefore provides a deeper
understanding of the associated chemical mechanisms that take place.
1.2 SCOPE OF THE TEXT
In Chapter 2, an outline of theory used throughout the monograph is provided. The
chapters that follow from this are presented primarily to provide descriptions of
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) previous mathematical and scientific investigations that have involved the use of
complex numbers. As each of these chapters evolved, new ideas inclined towards
research were conceived and duly incorporated at appropriate parts of the text. Three
research papers are central to Chapters 3 and 4. These are:
B. Riemann: Monatsberichte der Berliner Akademie, 1859, p671
H. von Mangoldt: Mathematische Annalen, 60, 1905, pi
P.P. Ewald: Annalen der Physik, 64, 1921, p253
Chapter 3 consists of a study of the application of complex analysis in number
theory with respect to Riemann's zêta function and is composed of three main areas.
The first of these is mainly introductory with some discussion of the functional
equation of the zeta function and line integration processes that are used to derive it.
Line integration is one of the first mathematical tricks that must be understood
Sec. 1.3] Riemann, zeta function 3
before other parts of Riemann's paper can be followed. Attention is then turned
towards work carried out by von Mangoldt approximately forty years after
Riemann's paper first appeared. This work describes a contour integration process
that provides important information on the quantity of roots of the zeta function.
More detail than available in von Mangoldt's original work is provided in Chapter 3
in order to clarify certain parts of the paper that might seem complicated at first
sight. Finally the roots of the zeta function are examined with the Euler-Maclaurin
summation technique. This technique is described in some depth, with details
provided on how it can be used to obtain information on the overall behaviour of
ζ(β). This section also includes an analysis of the distribution of roots, with frequent
referral to Haselgrove's tables.
Chapter 4 is concerned with an examination of the use of complex numbers within
theoretical physics. It consists of a study of Ewald's method with many analytical
calculations performed in order to facilitate conceptual understanding. The Ewald
method relies on complex algebra to provide a means of increasing the efficiency of
the calculation of energies within solid lattices. Chapter 4 is essentially a translation
of work written in German by Ewald in the 1920s, but with emphasis placed on the
use of graphical representations of equations to provide clearer descriptions. The
sections that follow provide some biographical information on Ewald, Riemann and
von Mangoldt.
1.3 G. F. B. RIEMANN AND THE ZETA FUNCTION
German mathematician Georg Friedrich Bernhard Riemann (1826-1866) enrolled at
Gottingen University in 1846 as a student of philology and theology and moved to
Berlin to study mathematics from 1847 to 1849. Dirichlet and Jacobi were amongst
some of the lecturers who were present during his time at Berlin. On completion of
his thesis in 1851, Riemann became assistant to W. Weber (1804-1891) again at
Gottingen, and by 1857 he held the position of Assistant Professor. These notes have
been taken from an article by Burkill (2002).
In Chapter 3, Riemann's work on the zeta function is studied and described with
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) reference to various texts that have been also been written on this topic. The zeta
function is an analytic function; analytic loosely defined as a complex variable
which is a function of another complex variable i.e. ζ(σ+ίί) where σ is a real
component; t the magnitude of the imaginary component, and ζ is the zeta function
which is another complex number that can be evaluated by a summation procedure
outlined in the introduction of Chapter 3.
Articles that concern the Riemann zeta function often include information on
an important hypothesis that he made within work which was published under the
heading "On the number of prime numbers less than a given magnitude" (Riemann,
1859). This hypothesis is presently considered one of the most important unsolved
problems in mathematics today and quite recently (2000) the Clay Institute in New
York offered $1 million for its solution (Yandell, 2002). Chapter 3 includes some
new information on the roots of the zeta function that appeared during a study of
4 Introduction [Ch. 1
Riemann's hypothesis and the asymptotic formula for the number of non-trivial
zeros.
1.4 STUDIES OF THE XI FUNCTION BY H. VON MANGOLDT
Contour integration of the xi function that is described in Riemann's memoir on the
zeta function was explained in papers by Hans von Mangoldt that were published
between 1895 and 1905. Beside teaching and carrying out research, von Mangoldt
(1854-1925), held senior administrative positions at two German universities during
his career. He published his first paper in 1875, concerned with previous work by
Gauss, at the age of only 21. His thesis was prepared in 1878 in Berlin with
supervision by K. Weierstrass (Knopp, 1927).
One of von Mangoldt's most important papers was produced in 1895 after a hiatus
associated with his publications of almost ten years. This work was presented in
Journal fur die Reine und Angewandte Mathematik and contains a thorough
investigation of Riemann's account of the zeta function. It is approximately fifty
pages long and divided into two sections. A fourteen page extract was published in
1894 and in 1896 a French translation of this extract was prepared by L. Laugel. The
first part of the work published in 1895 contains an account of the contour
integration of the xi function used to arrive at Riemann's equation that is suitable for
the determination of the number of roots of the zeta function on the critical line
(N(T)). A similar article was also published later in Mathematische Annalen in 1905
after the author had moved from Aachen to Danzig. The second part of the 1895
document contains a description of the derivation of Riemann's prime number
formula. This is the equation that actually provides some idea of how many prime
numbers are located below a certain magnitude. If RH is ever completely proven to
be correct then this formula will be important.
1.5 RECENT WORK ON THE ZETA FUNCTION
A large amount of computational work centred on Riemann's hypothesis has been
carried out, with computer programs able to provide extensive evaluations of
Riemann's zeta function. Important computational work related to the roots of C,(s)
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) has been published fairly recently by two separate research groups involved with
both the determination of quantities of roots on the critical line and theoretical
advances designed to locate non-trivial zeros at higher and higher values oft (van de
Lune, 1986 & Odlyzko, 1988). Other studies connected to the zeta function include
investigations of topics such as probabilities associated with prime numbers (Wagon,
1991), and random sequences linked to the location of zeros (Calude, 1997).
1.6 P. P. EWALD AND LATTICE SUMMATION
One of the aims of this text is to provide a link between the two worlds of
mathematics and theoretical physics. This has been attempted by including a chapter
on Ewald lattice summation which theoretical physicists sometimes discuss
alongside references to the study of zeta functions (Glasser, 1980 & Smith, 1981).
Paul P. Ewald was born in January 1888 in Berlin. He studied at Cambridge and
Sec. 1.6] P. Ewald, Lattice Summation 5
Gottingen Universities from 1905 to 1907 and obtained his doctorate at Munich
University in 1912. This was supervised by A. Sommerfeld with his thesis entitled
"Dispersion und Doppeibrechnung von Elektrongittern" (Ewald, 1916). It includes
complicated mathematics with the technique later recognised as Fourier
transformation (Ewald, 1962). From 1912-13 he was an assistant of Hubert's at
Gottingen and between 1914 and 1921 he worked with Sommerfeld once more at the
Institute for Theoretical Physics, Munich. A thesis on the dynamical theory of X-ray
diffraction was completed in 1917 and used to obtain a position as lecturer in
physics at Munich. In 1921 Ewald presented his summation procedure for the
calculation of electrostatic potentials within crystal lattices (Ewald, 1921). This
topic had received considerable attention since the work of Appell in 1884 which
was the first to position the concept of lattice sums within physics (Glasser, 1980).
Madelung obtained one of the first accurate estimates of the lattice energy of the
rocksalt structure in 1918, but his procedure unfortunately relied uncomplicated
geometric considerations, with summations over all lattice points and slow
convergence of Coulomb forces (Madelung, 1918). Ewald was able to overcome this
area of difficulty by a combination of procedures presented in Chapter 4 and
summarised as follows. First, the lattice sites were considered to take the form of
normal distributions; second, the idea of the reciprocal lattice was employed and
thirdly, a split integral was used during the summation process, resulting in
improved potential convergence.
From 1939-49 Ewald was a lecturer and later a professor of mathematical physics at
Queen's University of Belfast, Northern Ireland. He then moved to New York where
he worked as professor and head of a physics department at the Polytechnic Institute
of Brooklyn (1949-57).
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2
Theory
2.1 COMPLEX NUMBER ARITHMETIC
The following sections provide information on fundamental arithmetical procedures
that are used in any study involving complex numbers.
2.1.1 Addition and subtraction
For example, consider the following questions on the addition and subtraction of
complex numbers.
Example 2.1.1.1
(3 + 6i) + (2 + 5i) = 5+lli.
Example 2.1.1.2
(6 + 7i) - (3 - 2i) = 3 + 9i.
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These are straightforward procedures where the real and imaginary components are
either added or subtracted.
2.1.2 Multiplication
The following example demonstrates how complex numbers can be multiplied.
Example 2.1.2.1
(3 + 7i) x (4 + 5i) = 12 + 15i +28i - 35 = -23 + 43i.
Example 2.1.2.2
(6 - 5i) x (5 + 8i) = 30 + 48i - 25i + 40 = 70 + 23i.
8 Theory [Ch. 2
2.1.3 Division
The division of complex numbers is slightly more difficult owing to the necessary
use of a complex quotient for multiplication of the division in question. Consider the
following example.
Example 2.1.3.1
(3 + 5i) -H (2 + 2i)
To carry this calculation out it is important to multiply through by a
quotient that consists of the conjugate of 2 + 2i on both the numerator and
the denominator.
3 + 5/ 2-2/ 6-6/ + 10/ + 10 16 + 4/ „ .
-x = = = 2 + 0.5n/e.
2 + 2/ 2-2/ 4 + 4
Example 2.1.3.2
(4 +7i) + (3 - 9i)
4 + 7/ 3 + 9/ 12 + 36/ + 21/-63 -51 + 57/ ^.
n cn n
x = = = -0.57 + 0.63z.
3-9/ 3 + 9/ 9 + 81 90
2.2 ARGAND DIAGRAMS
Representation of complex numbers on Cartesian coordinates is often referred to in
Chapter 4. This process simply involves plotting the imaginary coordinate of the
complex number on the y-axis and the real component on the x-axis. The resulting
figure is known as an Argand diagram and an example of one of these is presented in
Fig. 2.2.1.
In this figure the complex number 2 + 3i can also be expressed as r(cos0 + isinO)
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where r is the modulus. This is often termed polar form or modulus-argument form.
The modulus is the length of the hypotenuse and is often expressed as a number
within brackets such as |4| or |25| for example. Θ is the angle that the hypotenuse
makes with the real axis.
2.3 EULER IDENTITIES
Another important aspect of complex number theory which is used frequently
throughout this account is concerned with the identities that Euler produced in the
1740s. These are stated as:
sin x = (e,x - e"ix)/2i cos x = (e,x + e"ix)/2
e'x = cos x + isin x e"'x = cos x - isin x.
Sec. 2.3] Euler Identities
Im A
2 + 3i
/
angle Θ 3
►
2 Re
Figure 2.2.1 : An example of an Argand diagram.
The use of e±,x is quite common when any work concerning complex numbers is
studied. Representation of e±lx by cosx ± isinx allows e±lx to be studied by Argand
diagram. This provides a clearer means of analysing e±lx. Two of the following
examples show how Argand diagrams can be constructed from Euler identities.
Example 2.3.1
π
In an Argand diagram, plot the complex number: e 4 .
é>4 = cos —π + /sin —π = 0.707 + 0.707/.
4 4
The diagram is shown in Fig. 2.3.1.
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Im
0.7
0.7 Re
Figure 2.3.1: Answer to Example 2.3.1.
10 Theory [Ch. 2
Example 2.3.2
Express e as a complex number.
e"421 = cos42° - isin42° = 0.743 - 0.669i.
Example 2.3.4
-\+—m
Plot e 5 on an Argand diagram with the help of an Euler identity.
, ' ■ ! ·
—H—τα . —m ,
e s -i 5 =o.368(cos 36° +i sin 36°) =0.298 + 0.216i.
=e e
Fig. 2.3.2.
Im
0.2
0.3 R
Figure 2.3.2: Answer to Example 2.3.4.
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Example 2.3.5
Express e"56' as a complex number.
e"561 = cos56° - isin56° = 0.559 - 0.829i.
2.4 POWERS AND LOGARITHMS
Throughout the study of the zeta function in Chapter 3, complex numbers are
frequently used as indices, and logarithms of complex numbers are analysed in some
instances as well.
2.4.1 Powers
If a complex number is abbreviated by s, and if any number x is raised to the power
s, it is possible to write: