Elements of Classical Analysis

Appendix A Elements of Classical Analysis A.1 Exotic Constants He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times. Gottfried Wilhelm Leibniz (1646–1716) The Euler–Mascheroni constant, γ, is the special constant defined by 1 1 γ = lim 1 + + ···+ − lnn = 0.57721 56649 01532 86060.... n→∞ 2 n In other words γ measures the amount by which the partial sum of the harmonic series differs from the logarithmic term lnn. The connection between 1 + 1/2 + 1/3 + ···+ 1/n and lnn was first established in 1735 by Euler [41], who used the notation C for it and stated that was worthy of serious consideration. It is worth mentioning that it was the Italian geometer Lorenzo Mascheroni (1750–1800) who actually introduced the symbol γ for the constant (although there is controversy about this claim) and also computed, though with error, the first 32 digits [36]. Unsurprisingly, establishing the existence of γ has attracted many proofs [65, pp. 69–73], [133, p. 235] to mention a few. However, a short and elegant proof is based on showing that the sequence (1 + 1/2 + 1/3 + ···+ 1/n − lnn)n≥1 is strictly decreasing and bounded below by 0. The Euler–Mascheroni constant, γ, considered to be the third important mathematical constant next to π and e, has appeared in a variety of mathematical formulae involving series, products, and integrals [43, pp. 28–32], [65, p. 109], [122, pp. 4–6], [123, pp. 13–22]. For an interesting survey paper on γ, as well as other reference material, the reader is referred to [34]. We mention that it is still an open problem to determine whether γ is a rational or an irrational number. O. Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical 253 Analysis, Problem Books in Mathematics, DOI 10.1007/978-1-4614-6762-5, © Springer Science+Business Media New York 2013 254 A Elements of Classical Analysis The nth harmonic number, Hn, is defined for n ≥ 1by 1 1 H = 1 + + ···+ . n 2 n In other texts, it is also defined as the partial sum of the harmonic series ∞ 1 n 1 ∑ = lim ∑ = ∞. n→∞ k=1 k k=1 k The nth harmonic number verifies the recurrence formula Hn = Hn−1 + 1/n for n ≥ 2 with H1 = 1 which can be used to establish the generating function ( − ) ∞ −ln 1 x = n, − ≤ < . − ∑ Hnx 1 x 1 1 x n=1 Significantly, one of the most important properties on the nth harmonic number refers to its divergence. The first proof of the divergence of the harmonic series is attributed, surprisingly, to Nicholas Oresme (1323–1382), one of the greatest philosophers of the Middle Ages, and similar proofs were discovered in later centuries by the Bernoulli brothers. Even Euler proved this result by evaluating an improper integral in two different ways [65, p. 23]. Another curious property is that Hn is nonintegral. This means that even though Hn increases without bound, it avoids all integers, apart from n = 1. The nth harmonic number has been involved in a variety of mathematical results: from summation formulae involving the binomial coefficients to a nice formula that expresses the nth harmonic number in terms of the unsigned Stirling numbers of the first kind [13]. A detailed history on the nth harmonic number as well as its connection to γ can be found in Havil’s book [65]. The Glaisher–Kinkelin constant, A, is defined by the limit n 2 2 A = lim n−n /2−n/2−1/12en /4 ∏ kk = 1.28242 71291 00622 63687.... n→∞ k=1 Equivalently, n n2 n 1 n2 lnA = lim ∑ k lnk − + + lnn + . n→∞ k=1 2 2 12 4 Another remarkable formula, involving factorials, in which A appears is the following: 1 1! · 2!···(n − 1)! e 12 lim = . →∞ − 3 n2 1 n 1 n2− 1 n e 4 (2π) 2 n 2 12 A These formulae are due to Kinkelin [72], Jeffery [71], and Glaisher [59–61]. A.1 Exotic Constants 255 √ The constant A, which plays the same role as 2π plays in Stirling’s formula, has the following closed-form expression [43, p. 135]: 1 ζ (2) ln(2π)+γ A = exp − ζ (−1) = exp − + , 12 2π2 12 where ζ denotes the derivative of the Riemann zeta function and γ is the Euler– Mascheroni constant. Many beautiful formulae including A exist in the literature, from infinite products and definite integrals [43, pp. 135–136] to the evaluation of infinite series involving Riemann zeta function [122]. Other formulae involving infinite series and products, believed to be new in the literature, in which A appears, are recorded as problems in this book. The Stieltjes constants, γn, are the special constants defined by m (lnk)n (lnm)n+1 γn = lim ∑ − , m→∞ + k=1 k n 1 and, in particular, γ0 = γ, the Euler–Mascheroni constant. They occur in the Laurent expansion of the Riemann zeta function in a neighborhood of its simple pole at z = 1(see[98, Entry 25.2.4, p. 602]): ∞ (− )n ζ( )= 1 + 1 γ ( − )n, ℜ( ) > . s − ∑ n s 1 s 0 s 1 n=0 n! Interesting properties related to the sign of the Stieltjes constants as well as an open problem involving their magnitudes are recorded in [43, pp. 166–167]. The logarithmic constants, δn, are defined by m m n n 1 n n (n) δn = lim ∑ ln k − ln xdx − ln m =(−1) (ζ (0)+n!). m→∞ k=1 1 2 In particular δ0 = 1/2, δ1 = 1/2ln(2π) − 1. Sitaramachandrarao [118] proved that they appear in the Laurent expansion of the Riemann zeta function at the origin rather than at unity ∞ (− )n ζ( )= 1 + 1 δ n. z − ∑ nz z 1 n=0 n! We called these constants logarithmic for simplicity and because of the logarithmic terms involved in their definition. The constants, which are also discussed in [43, p. 168], have been used in Problem 3.98 in order to evaluate a double logarithmic series. 256 A Elements of Classical Analysis A.2 Special Functions Euler’s integral appears everywhere and is inextricably bound to a host of special functions. Its frequency and simplicity make it fundamental. Philip J. Davis Euler’s Gamma function, which extends factorials to the set of complex numbers, is a function of a complex variable defined by ∞ − − Γ(z)= xz 1e xdx, ℜ(z) > 0. 0 When ℜ(z) ≤ 0, the Gamma function is defined by analytic continuation. It is a meromorphic function with no zeros and with simple poles of residue (−1)n/n!at z = −n. Historically, the Gamma function was first defined by Euler,1 as the limit of a product 1 · 2···(n − 1) Γ(z)=lim nz, n→∞ z(z + 1)(z + 2)···(z + n − 1) ∞ z−1 −x from which the infinite integral 0 x e dx can be derived. Weierstrass defined the Gamma function as an infinite product2 ∞ 1 γ z − z = z + n . Γ( ) ze ∏ 1 e z n=1 n The interesting and exotic Gamma function has attracted the interest of famous mathematicians such us Stirling, Euler, Weierstrass, Legendre, and Schlomilch.¨ Even Gauss worked on it: his paper on Gamma function was a part of a larger work on hypergeometric series published in 1813. There are lots of interesting properties that are satisfied by the Gamma function; for example, it satisfies the difference equation Γ(z + 1)=z · Γ(z) and Euler’s reflection formula 1Euler did not introduce a notation for the Gamma function. The notation Γ was introduced by Legendre, in 1814, while Gauss used Π. 2Weierstrass defined the Gamma function by this product in 1856. In fact, it was the German Γ( )= −γm/ ∞ ( + / )−1 m/k mathematician Schlomilch¨ who discovered the formula m e m∏k=1 1 m k e even earlier, in 1843 (see [110, p. 457]). A.2 Special Functions 257 π Γ(z) · Γ(1 − z)= , z = 0,±1,±2,... sinπz to mention a few. The Gamma function is one of the basic special functions that plays a significant part in the development of the theory of infinite products and the calculation of certain infinite series and integrals. More information, as well as historical accounts on the Gamma function can be found in [110, pp. 444–475] and [133, pp. 235–264]. The Beta function is defined by 1 − − B(a,b)= xa 1(1 − x)b 1dx, ℜ(a) > 0, ℜ(b) > 0. 0 In his work on Gamma function, Euler discovered the important theorem that B(a,b)· Γ(a + b)=Γ(a)· Γ(b), which connects the Beta with the Gamma function. Other integral representations of the Beta function as well as related properties and formulae can be found in [110]and[123]. The Riemann zeta function is defined by ∞ ζ( )= 1 = + 1 + ···+ 1 + ···, ℜ( ) > . z ∑ z 1 z z z 1 n=1 n 2 n Elsewhere ζ(z) is defined by analytic continuation. It is a meromorphic function whose only singularity in C is a simple pole as z = 1, with residue 1. Although the function was known to Euler, its most remarkable properties were not discovered before Riemann3 who discussed it in his memoir on prime numbers. The Riemann zeta function is perhaps the most important special function in mathematics that arises in lots of formulae involving integrals and series ([98, pp. 602–606], [122]) and is intimately related to deep results surrounding the Prime Number Theorem [39]. While many of the properties of this function have been investigated, there remain important fundamental conjectures, most notably being the Riemann hypothesis which states that all nontrivial zeros of ζ lie on the vertical line ℜ(z)=1/2.

Load more