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 √-1 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 imaginary numbers was recorded. Worthy of note are contributions made by scholars such as Diophantus 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 i is representative of √-1 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 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 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 Chapter 4. These are:

B. Riemann: Monatsberichte der Berliner Akademie, 1859, p671

H. von Mangoldt: Mathematische Annalen, 60, 1905, p1

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 zeta 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 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(s). 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 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 + it) 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 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...