Search

This chapter is devoted entirely to the theory and application of infinite products, and as a consolation prize, we also talk about partial fractions.

Suppose we are interested in an infinite sequence of numbers \(\left (a_{k}\right )_{k=0}^{\infty }\) that arises naturally from geometric problems, or from recursively defined problems. Is there a “good formula” for a k as a function of k? Are there identities involving various a k ’s? Embedding this sequence into the generating function \(F(z) =\sum _{k\geq 0}a_{k}\,z^{k}\) allows us to retrieve answers to the questions above in a surprisingly quick and elegant way. We may think of F(z) as lifting our sequence a k from its discrete setting into the continuous world of functions. We introduce techniques for working with generating functions, and we use them to shed light on the Frobenius coin-exchange problem: Given relatively prime positive integers \(a_{1},a_{2},\ldots,a_{n}\), what is the largest integer that cannot be written as a nonnegative integral linear combination of \(a_{1},a_{2}\ldots,a_{n}\)?

We encountered Dedekind sums in our study of finite Fourier analysis in Chapter 7, and we became intimately acquainted with their siblings in our study of the coin-exchange problem in Chapter 1 They have one shortcoming, however (which we shall remove): the definition of s(a, b) requires us to sum over b terms, which is rather slow when b = 2100, for example. Luckily, there is a magical reciprocity law for the Dedekind sum s(a, b) that allows us to compute it in roughly \(\log _{2}(b) = 100\) steps in this example. This is the kind of magic that saves the day when we try to enumerate lattice points in integral polytopes of dimension d ≤ 4. In this chapter, we focus on the computational-complexity issues that arise when we try to compute Dedekind sums explicitly. In many ways, the Dedekind sums extend the notion of the greatest common divisor of two integers.

As I mentioned in the Preface, in 1697 John Bernoulli evaluated the exotic-looking integral $$ {\displaystyle {\int}_0^1{\mathrm{x}}^{\mathrm{x}}\mathrm{dx}}. $$

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers \(\{F_{n}\}_{n=1}^{\infty }\); Lekkerkerker proved that the average number of summands for integers in [F n , F n+1) is \(n/(\varphi ^{2} + 1)\), with φ the golden mean. Interestingly, the higher moments seem to have been ignored. We discuss the proof that the distribution of the number of summands converges to a Gaussian as n → ∞, and comment on generalizations to related decompositions. For example, every integer can be written uniquely as a sum of the ± F n ’s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely \(-(21 - 2\varphi )/(29 + 2\varphi ) \approx -0.551058\).

The material in this chapter was evidently intended for the conclusion of Ramanujan’s paper, Some formulae in the analytic theory of numbers. We see how partial fraction expansions were used by Ramanujan to derive many of his identities for infinite series, including his famous formula for \(\zeta (2n + 1)\).

This chapter is evidently a preliminary version of Ramanujan’s paper on certain infinite products.

In this chapter, we discuss the general theory of quadratic forms and the notion of congruence and its invariants, as well as the applications of the theory to the analysis of special forms. We will focus primarily on quadratic forms induced by rational functions, most notably the Hankel and Bezout forms, because of their connection to system-theoretic problems such as stability and signature symmetric realizations. These forms use as their data different representations of rational functions, namely power series and coprime factorizations respectively. But we will discuss also the partial fraction representation in relation to the computation of the Cauchy index of a rational function and the proof of the Hermite–Hurwitz theorem and the continued fraction representation as a tool in the computation of signatures of Hankel matrices as well as in the problem of Hankel matrix inversion. Thus, different representations of rational functions, i.e., different encodings of the information carried by a rational function, provide efficient starting points for different methods. The results obtained for rational functions will be applied to root-location problems for polynomials in the next chapter.

In this chapter, the word D–module is used to denote a module over one of the two Weyl algebras W(V),W(U) of differential operators with polynomial coefficients on V,U respectively. The purpose of this chapter is to determine an expansion in partial fractions of the regular functions on the complement of a hyperplane arrangement. This is essentially the theory of Brion–Vergne (cf. [28], [7]). We do it using the D–module structure of the algebra of regular functions. Finally, by inverse Laplace transform all this is interpreted as a calculus on the corresponding distributions.

In Section 1 we derive the Laurent decomposition of a function that is analytic on an annulus, and in Section 2 we use the Laurent decomposition on a punctured disk to study isolated singularities of analytic functions. We classify these as removable singularities, essential singularities, or poles, and we characterize each type of singularity. In Section 3 we define isolated singularities at ∞, and in Section 4 we derive the partial fractions decomposition of a rational function. In Sections 5 and 6 we use the Laurent decomposition to study periodic functions and we relate Laurent series to Fourier series. Sections 5 and 6 can be omitted at first reading.

In the last four chapters we used the so-called time domain analysis. In this approach we investigate difference equations as they are, that is, without transforming them into another domain. We either find solutions of the difference equations or provide information about their qualitative behavior.

The main topic of this chapter is the structure of solutions of a linear system $$ \frac{{d\vec y}}{{dt}} = A(t)\vec y + \vec b(t), $$ (LP) where entries of the n × n matrix A(t) are complex-valued (i.e.,ℂ-valued) continuous functions of a real independent variable t, and the ℂn-valued function \(\mathop{{{\text{ }}b}}\limits^{ \to } (t)\) is continuous in t. The existence and uniqueness of solutions of problem (LP) were given by Theorem I-3-5. In §IV-1, we explain some basic results concerning n x n matrices whose entries are complex numbers. In particular, we explain the S-N decomposition (or the Jordan-Chevalley decomposition) of a matrix (cf. Definition IV-1-12; also see [Bou, Chapter 7], [HirS, Chapter 6], and [Hum, pp. 17-18]). The S-N decomposition is equivalent to the block-diagonalization which separates distinct eigenvalues. It is simpler than the Jordan canonical form. The basic tools for achieving this decomposition are the Cayley-Hamilton theorem (cf. Theorem IV-1-5) and the partial fraction decomposition of reciprocal of the characteristic polynomial. It is relatively easy to obtain this decomposition with an elementary calculation if all eigenvalues of a given matrix are known (cf. Examples IV-1-18 and IV-1-19). In §IV-2, we explain the general aspect of linear homogeneous systems. Homogeneous systems with constant coefficients are treated in §IV-3. More precisely speaking, we define e tA and discuss its properties. In §IV-4, we explain the structure of solutions of a homogeneous system with periodic coefficients. The main result is the Floquet theorem (cf. Theorem IV-4-1 and [F1]).

The negative integers, fractions, irrationals and imaginary numbers were all introduced in order to supply solutions to polynomial equations. Do we need more numbers? Is there a polynomial equation with complex coefficients with a root that is not a complex number? The answer is no, and the fact that no new numbers are required is called the Fundamental Theorem of Algebra. it was first stated by Albert Girard (1595-1632) in 1629, and proofs were given by Jean d’Alembert (1717-1783) and Carl Friedrich Gauss (1777-1855). However, John Stillwell [1989] argues that there is a flaw in Gauss’s proof, and the first rigorous proof was given only after Weierstrass established the basic properties of continuous functions.

We have seen that if F is a field, every nonconstant polynomial in F[x] factors uniquely (up to the order of the factors) into the product of irreducible polynomials. Irreducible polynomials relate to all polynomials in the same way that primes do to all natural numbers. Thus one naturally asks: Which polynomials are irreducible? and How does one factor a given polynomial into a product of irreducible polynomials?

Ramanujan evidently had an affinity for partial fraction expansions, which can be found in several places in his notebooks. The heaviest concentrations lie in Chapters 14 and 18 and in the unorganized pages at the end of the second notebook. See our books [4] and [6] for accounts of the material in Chapters 14 and 18, respectively. In this chapter, we prove the 15 partial fraction decompositions found in the unorganized pages of the second notebook.

This chapter develops concisely the most important discrete-time analogs of Chapter 7 on continuous-time stability: I/O stability, state related stability concepts and responses to q-periodic inputs. Certain specific details such as the partial fraction expansion (7d.1.56) are added for clarity.

In Chapter 18, Ramanujan continues his development of the theory of elliptic functions begun in Chapter 16 with the theory of theta-functions and continued in Chapter 17 with an introduction to elliptic integrals and the compilation of a large catalog of series that can be evaluated in terms of elliptic function parameters. This chapter contains further series identities depending on the theory of elliptic functions. Such results are considerably fewer in number here than in Chapter 17 and generally are more difficult to prove. In particular, see Sections 4-7.

The purpose of this chapter is to teach you certain basic tricks to find indefinite integrals. It is of course easier to look up integral tables, but you should have a minimum of training in standard techniques.

Besides the basic methods of integration associated with reversing the differentiation rules, there are special methods for integrands of particular forms. Using these methods, we can solve some interesting length and area problems.

In the last three chapters we have accumulated a rather large number of formulas for both differentiation and integration. It is the purpose of this chapter to bring these results together, and to present ways by which these results can be extended. It would therefore be helpful to place the results obtained so far in tabular form for easy reference. We list here the results for both derivatives and integrals, pairing them off where this is appropriate. The differential can be easily obtained from the derivative, and for that reason will not be listed in this table.