Mathematical Preliminaries

This chapter presents mathematical concepts that are used throughout the book; much of this material should be already known to well-prepared students. Topics covered: Infinite series, rules for determining the convergence of series, the binomial expansion, vectors (including addition, multiplication by a scalar, dot product), complex numbers and functions, Euler’s formula (complex exponential), multiple-valued functions, partial, chain rule, conditions for extrema of functions of several variables, multiple integrals and their expression in different coordinate systems, and the Dirac delta function. Also included is the method of proof of theorems using mathematical induction.

Keywords: infinite series, binomial expansion, vectors, dot product, complex functions, multiple-valued functions, partial differentiation, multiple integrals, Dirac delta function, mathematical induction

This introductory chapter surveys a number of mathematical techniques that are needed throughout the book. Some of the topics (e.g., complex variables) are treated in more detail in later chapters, and the short survey of special functions in this chapter is supplemented by extensive later discussion of those of particular importance in physics (e.g., Bessel functions). A later chapter on miscellaneous mathematical topics deals with material requiring more background than is assumed at this point. The reader may note that the Additional Readings at the end of this chapter include a number of general references on mathematical methods, some of which are more advanced or comprehensive than the material to be found in this book.

1.1 Infinite Series

Perhaps the most widely used technique in the physicist’s toolbox is the use of infinite series (i.e., sums consisting formally of an infinite number of terms) to represent functions, to bring them to forms facilitating further analysis, or even as a prelude to numerical evaluation. The acquisition of skill in creating and manipulating series expansions is therefore an absolutely essential part of the training of one who seeks competence in the mathematical methods of physics, and it is therefore the first topic in this text. An important part of this skill set is the ability to recognize the functions represented by commonly encountered expansions, and it is also of importance to understand issues related to the convergence of infinite series.

Fundamental Concepts

The usual way of assigning a meaning to the sum of an infinite number of terms is by introducing the notion of partial sums. If we have an infinite sequence of terms u1, u2, u3, u4, u5,…,we define the i th partial sum as

i=∑n=1iun. (1.1)

This is a finite summation and offers no difficulties. If the partial sums si converge to a finite limit as →∞,

i→∞si=S, (1.2)

the infinite series n=1∞un is said to be convergent and to have the value S. Note that we define the infinite series as equal to S and that a necessary condition for convergence to a limit is that n→∞un=0. This condition, however, is not sufficient to guarantee convergence.

Sometimes it is convenient to apply the condition in Eq. (1.2) in a form called the Cauchy criterion, namely that for each >0 there is a fixed number N such that sj−si|<ε for all i and j greater than N. This means that the partial sums must cluster together as we move far out in the sequence.

Some series diverge, meaning that the sequence of partial sums approaches ∞; others may have partial sums that oscillate between two values, as for example,

n=1∞un=1−1+1−1+1−⋯−(−1)n+⋯.

This series does not converge to a limit, and can be called oscillatory. Often the term divergent is extended to include oscillatory series as well. It is important to be able to determine whether, or under what conditions, a series we would like to use is convergent.

We now turn to a more detailed study of the convergence and divergence of series, considering here series of positive terms. Series with terms of both signs are treated later.

Comparison Test

If term by term a series of terms un satisfies ≤un≤an, where the an form a convergent series, then the series nun is also convergent. Letting si and sj be partial sums of the u series, with j > i, the difference j−si is n=i+1jun, and this is smaller than the corresponding quantity for the a series, thereby proving convergence. A similar argument shows that if term by term a series of terms vn satisfies ≤bn≤vn, where the bn form a divergent series, then nvn is also divergent.

For the convergent series an we already have the geometric series, whereas the harmonic series will serve as the divergent comparison series bn. As other series are identified as either convergent or divergent, they may also be used as the known series for comparison tests.

Cauchy Root Test

If an)1∕n≤r<1 for all sufficiently large n, with r independent of n, then nan is convergent. If an)1∕n≥1 for all sufficiently large n, then nan is divergent.

The language of this test emphasizes an important point: The convergence or divergence of a series depends entirely on what happens for large n. Relative to convergence, it is the behavior in the large-n limit that matters.

The first part of this test is verified easily by raising an)1∕n to the n th power. We get

n≤rn<1.

Since rn is just the n th term in a convergent geometric series, nan is convergent by the comparison test. Conversely, if an)1∕n≥1, then n≥1 and the series must diverge. This root test is particularly useful in establishing the properties of power series (Section 1.2).

D’Alembert (or Cauchy) Ratio Test

If...