360 SERIES 8.4.2-2. Fourier expansion of odd and even functions. 1 ◦ .Letf(x) be an even function, i.e., f(x)=f(–x). Then the Fourier expansion of f(x) on the interval (–l, l) has the form of the cosine Fourier series: f(x)= a 0 2 + ∞  n=1 a n cos nπx l , where the Fourier coefficients have the form a n = 2 l  l 0 f(x)cos nπx l dx (b n = 0). 2 ◦ .Letf(x) be an odd function, i.e., f(x)=–f(–x). Then the Fourier expansion of f(x) on the interval (–l, l) has the form of the sine Fourier series: f(x)= ∞  n=1 b n sin nπx l , where the Fourier coefficients have the form b n = 2 l  l 0 f(x)sin nπx l dx (a n = 0). Example. Let us find the Fourier expansion of the function f (x)=x on the interval (–π, π). Taking l = π and f(x)=x in the formula for the Fourier coefficients and integrating by parts, we obtain b n = 2 π  π 0 x sin(nx) dx = 2 π  – 1 n x cos(nx)    π 0 + 1 n  π 0 cos(nx) dx  =– 2 n cos(nπ)=(–1) n+1 2 n . Therefore, the Fourier expansion of f (x)=x has the form f(x)=2 ∞  n=1 (–1) n+1 sin(nx) n (–π < x < π). 3 ◦ .Iff (x)isdefined on the interval (0, l) and satisfies the Dirichlet conditions, it can be represented by the cosine Fourier series, as well as the sine Fourier series (with the help of the above formulas).* Both series on the interval (0, l) give the values of f(x) at points of its continuity and the value 1 2 [f(x 0 + 0)+f(x 0 – 0)] at points of its discontinuity; outside the interval (0, l), these two series represent different functions. 8.4.2-3. Fourier series in complex form. The complex Fourier expansion of a function f(x)onaninterval(–l, l)hastheform f(x)= ∞  n=–∞ c n e iω n x , where ω n = nπ l , c n = 1 2l  l –l f(x)e –iω n x dx; n = 0, 1, 2, The expressions e iω n x are called harmonics, the coefficients c n are complex amplitudes, ω n are wave numbers of the function f(x), and the set of all wave numbers {ω n } is called the discrete spectrum of the function. * The cosine Fourier expansion of f(x) on the interval (0, l) corresponds to the extension of f(x)tothe interval (–l, 0) as an even function: f(–x)=f(x). The sine Fourier expansion of f(x)on(0, l) corresponds to the extension of f(x) to the interval (–l, 0) as an odd function: f(–x)=–f(x). 8.4. FOURIER SERIES 361 8.4.3. Criteria of Uniform and Mean-Square Convergence of Fourier Series 8.4.3-1. Criteria of uniform convergence of Fourier series. LIPSCHITZ CRITERION. The Fourier series of a function f(x) converges uniformly to that function on an interval [–l, l] if on a wider interval [–L, L](–L <–l < l < L) the following inequality holds: |f(x 1 )–f(x 2 )| ≤ K|x 1 – x 2 | σ for all x 1 , x 2 [L, L], where K and σ are constants, 0 < σ ≤ 1 . Corollary. The Fourier series of a continuous function f(x) uniformly converges to that function on an interval [–l, l] if on a wider interval the function f (x) has a bounded derivative f  (x). D IRICHLET–JORDAN CRITERION. Suppose that on an interval (–l, l) [–L, L] , a contin- uous function f(x) has bounded variation (i.e., can be represented as the difference of two monotonically increasing functions). Then its Fourier series is uniformly convergent to that function on the interval (–l, l) . For any continuously differentiable 2l-periodic function f(x), its Fourier series [defined by formulas (8.4.2.1)–(8.4.2.2)] is uniformly convergent to f(x). 8.4.3-2. Fourier series of square-integrable functions. Parseval identity. 1 ◦ . For a continuous 2π-periodic function f(x), its Fourier series (8.4.1.1)–(8.4.1.2) con- verges to f(x) in mean square, i.e.,  π –π   f(x)–f n (x)   2 dx → 0 as n →∞, where f n (x)= 1 2 a 0 + n  k=1 (a k cos kx + b k sin kx) is a partial sum of the Fourier series. 2 ◦ .Iff (x) is integrable on the segment [–π, π] and the integral  π –π |f(x)| 2 dx exists as an improper integral with finitely many singularities, then the Fourier series (8.4.1.1)–(8.4.1.2) is mean-square convergent to f (x). 3 ◦ .Letf (x) L 2 [–π, π] be a square-integrable function on the segment [–π, π]. Then its Fourier series (8.4.1.1)–(8.4.1.2) is mean-square convergent to f(x), and the Parseval identity holds: a 2 0 2 + ∞  n=1 (a 2 n + b 2 n )= 1 π  π –π f 2 (x) dx, where a n , b n are defined by (8.4.1.2). Note that the functions considered in Items 1 ◦ and 2 ◦ belong to L 2 [–π, π]. 362 SERIES 8.4.4. Summation Formulas for Trigonometric Series 8.4.4-1. Summation of trigonometric series with the help of Laplace transforms. When finding sums of trigonometric series, the following formulas may be useful: ∞  n=1 F (n)sin(nx)= 1 2 sin x  ∞ 0 f(t) dt cosh t –cosx , 0 ≤ x ≤ π; ∞  n=1 F (n)cos(nx)= 1 2  ∞ 0 cos x – e –t cosh t –cosx f(t) dt, 0 ≤ x ≤ π, where F (x)=  ∞ 0 f(t)e –xt dt. For specific F(x), the corresponding functions f (t) can be found in tables of inverse Laplace transforms (see Section T3.2). 8.4.4-2. Summation of series with the help of functions of complex variable. Suppose that trigonometric series a 0 2 + ∞  n=1 a n cos(nx)=f 1 (x), (8.4.4.1) ∞  n=1 a n sin(nx)=f 2 (x)(8.4.4.2) have positive (nonnegative) coefficients a n and the auxiliary series ∞  n=1 a n n is convergent. Then the series (8.4.4.1) and (8.4.4.2) are Fourier series representing continuous functions. In order to find the sums f 1 (x)andf 2 (x) of the series (8.4.4.1) and (8.4.4.2), it is sometimes possible to use functions of complex variable. Let ϕ(z)= a 0 2 + ∞  n=1 a n z n be the sum of a series that is convergent in the circle |z| < 1. If lim |z|→1 ϕ(z) = lim r→1 ϕ(re ix )= ϕ(e ix ), then f 1 (x)+if 2 (x)=ϕ(e ix ). Thus, after separating the real and the imaginary parts of the function ϕ(e ix ), we get f 1 (x)=Re[ϕ(e ix )], f 2 (x)=Im[ϕ(e ix )]. Example. Let us sum up the series 1 + ∞  n=1 cos(nx) n! = f 1 (x), ∞  n=1 sin(nx) n! = f 2 (x). 8.5. ASYMPTOTIC SERIES 363 Taking ϕ(z)=1 + ∞  n=1 z n n! = e z and representing z in exponential form z = re ix , let us pass to the limit as r → 1.Wefind that exp(e ix )=e cos x+i sin x = e cos x [cos(sin x)+i sin(sinx)] = 1 + ∞  n=1 (cos x + i sin x) n n! . Using the Moivre formula (cos x + i sin x) n =cos(nx)+i sin(nx), we obtain e cos x cos(sin x)+ie cos x sin(sin x)=1 + ∞  n=1 cos(nx) n! + i ∞  n=1 sin(nx) n! . It follows that 1 + ∞  n=1 cos(nx) n! = e cos x cos(sin x), ∞  n=1 sin(nx) n! = e cos x sin(sin x). 8.5. Asymptotic Series 8.5.1. Asymptotic Series of Poincar ´ e Type. Formulas for the Coefficients 8.5.1-1. Definition of asymptotic series. Illustrative example. Suppose that for large x the function f(x) can be represented in the form f(x)= n  k=0 a k x k + o  x –n  as x →∞.(8.5.1.1) Then one writes f(x) ∼ ∞  k=0 a k x k .(8.5.1.2) The infinite series on the right-hand side is called an asymptotic series of Poincar ´ e’s type (or an asymptotic expansion) of the function f(x). Asymptotic series may happen to be convergent or divergent. Example. Consider the function defined by the integral f(x)=  ∞ x e x–t dt t . Repeated integration by parts yields f(x)= 1 x – 1 x 2 + 2! x 3 – ···+(–1) n–1 (n – 1)! x n + R n (x), (8.5.1.3) where the remainder is defined by R n (x)=(–1) n n!  ∞ x e x–t t n+1 dt. The following estimate holds: |R n (x)| = n!  ∞ x e x–t t n+1 dt < n! 1 x n+1  ∞ x e x–t dt =– n! x n+1 e x–t    t=∞ t=x = n! x n+1 . 364 SERIES Therefore, R n (x)=o  x –n  and we can write f(x) ∼ 1 x – 1 x 2 + 2! x 3 – ···+(–1) n–1 (n – 1)! x n + ···. This asymptotic series is divergent, since for any fixed x we have lim n→∞ |a n | = lim n→∞ (n – 1)! x n = ∞. The motivation of using divergent asymptotic series of the form (8.5.1.2) is that a finite sum of the series (8.5.1.1) provides a good approximation for the given function for x →∞. For a fixed n, the accuracy of the approximations increases with the growth of x. 8.5.1-2. Uniqueness of an asymptotic series representing a function. 1. For a function f (x) that admits the representation by asymptotic series (8.5.1.2), such a representation is unique and the coefficients a n of the series are determined by a n = lim x→∞ x n  f(x)– n–1  k=0 a k x –k  . 2. There exist functions that are not identically equal to zero, but having all coefficients a n in their asymptotic expansion (8.5.1.2) equal to zero. Such a function is called asymptotic zero. The class of asymptotic zeroes includes any function f(x) such that for large x and any positive integer n the following estimate holds: f(x)=o(x –n ). For instance, the functions f (x)=e –x , f (x)=xe –2x , f (x)=exp(–x 2 ) belong to this class. Adding such functions to the left-hand side of the asymptotic expansion (8.5.1.2) does not change its right-hand side. C OROLLARY. The coefficients of the asymptotic expansion of a function f(x) do not uniquely determine the function. Remark. Addition of a suitable asymptotic zero can be used, in some cases, for the construction of approximations that not only correctly describe the behavior of a function for large x, but give fairly accurate numerical results for moderate values of x (i.e., x = O(1); sometimes even for small x). This method is often used in applications, for instance in engineering, to obtain formulas applicable in a wide range of some characteristic parameter. 8.5.2. Operations with Asymptotic Series 8.5.2-1. Addition, subtraction, multiplication, and division of asymptotic series. 1. Asymptotic expansions of functions f(x) ∼ ∞  n=0 a n x –n and g(x) ∼ ∞  n=0 b n x –n admit term-by-term addition and subtraction: f(x) g(x) ∼ ∞  n=0 a n x –n ∞  n=0 b n x –n = ∞  n=0 (a n b n )x –n . 8.5. ASYMPTOTIC SERIES 365 2. Asymptotic expansions f(x) ∼ ∞  n=0 a n x –n and g(x) ∼ ∞  n=0 b n x –n can be formally multiplied, according to the Cauchy rule: f(x)g(x) ∼  ∞  n=0 a n x –n  ∞  n=0 b n x –n  = ∞  n=0 c n x –n , c n = n  k=0 a k b n–k . 3. Asymptotic expansion f(x)∼ ∞  n=0 a n x –n can beformally divided by g(x)∼ ∞  n=0 b n x –n , provided that b 0 ≠ 0. The coefficients of the resulting asymptotic expansion h(x) ∼ ∞  n=0 b n x –n are found by the method of indefinite coefficients with the help of the relation f(x)=g(x)h(x): the corresponding expansions should be inserted into this relation and then similar terms gathered. 8.5.2-2. Composition of series. Integration of asymptotic series. 1. Suppose that in a neighborhood of y = 0, the function g(y) can be represented by a convergent power series, g(y)= ∞  m=0 b m y m ,andforx →∞, the function f (x) can be represented as an asymptotic series f(x) ∼ ∞  n=1 a n x –n with a 0 = 0. Then, for large enough x, the composite function g  f(x)  = ∞  m=0 b m [f(x)] m (8.5.2.1) makes sense and admits an asymptotic expansion, which can be obtained from (8.5.2.1) if f(x) is replaced by its asymptotic expansion, after which similar terms should be gathered. 2. Suppose that the asymptotic expansion of a function f(x) starts with a 2 x –2 [i.e., a 0 = a 1 = 0 in (8.5.1.2)]. Then this asymptotic expansion admits term-by-term integration from x to ∞,  ∞ x f(x) dx ∼  ∞ x  ∞  n=2 a n x n  dx = ∞  n=2  ∞ x a n x n dx = ∞  n=2 a n (n – 1)x n–1 . 3. Term-by-term differentiation of asymptotic series is impossible, in general. Example. To illustrate the last statement, consider the function f(x)=e –x sin(e x ). For all n,wehave lim x→∞ x n f(x)=0, and therefore the asymptotic expansion of this function has zero coefficients, f(x) ∼ 0. Term-by-term differentiation of this relation also yields zero, f  (x) ∼ 0. On the other hand, the derivative of this function, f  (x)=–e –x sin(e x )+cos(e x ), admits no asymptotic expansion at all, since there is no limit lim x→∞ f  (x)! 366 SERIES References for Chapter 8 Brannan, D., A First Course in Mathematical Analysis, Cambridge University Press, Cambridge, 2006. Bromwich,T.J.I.,Introduction to the Theory of Infinite Series, American Mathematical Society, Providence, Rhode Island, 2005. Bronshtein, I. N. and Semendyayev, K. A., Handbook of Mathematics, 4th Edition, Springer-Verlag, Berlin, 2004. Danilov, V. L., Ivanova, A. N., et al., Mathematical Analysis (Functions, Limits, Series, Continued Fractions) [in Russian], Fizmatlit Publishers, Moscow, 1961. Davis, H. F., Fourier Series and Orthogonal Functions, Dover Publications, New York, 1989. Fedoryuk, M. V. Asymptotics, Integrals, Series [in Russian], Nauka, Moscow, 1987. Fichtenholz, G. M., Functional Series, Taylor & Francis, London, 1970. Fikhtengol’ts (Fichtenholz), G. M., A Course in Differential and Integral Calculus, Vols. 2 and 3 [in Russian], Nauka Publishers, Moscow, 1969. Gradshteyn,I.S.andRyzhik,I.M.,Tables of Integrals, Series, and Products, 6th Edition, Academic Press, New York, 2000. Hansen,E.R.,A Table of Series and Products, Printice Hall, Englewood Cliffs, London, 1975. Hirschman, I., Infinite Series, Greenwood Press, New York, 1978. Hyslop, J. M., Infinite Series, Dover Publications, New York, 2006. Jolley, L. B. W., Summation of Series, Dover Publications, New York, 1961. Knopp, K., Infinite Sequences and Series, Dover Publications, New York, 1956. Knopp, K., Theory and Application of Infinite Series, Dover Publications, New York, 1990. Mangulis, V., Handbook of Series for Scientists and Engineers, Academic Press, New York, 1965. Pinkus, A. and Zafrany, S., Fourier Series and Integral Transforms, Cambridge University Press, Cambridge, 1997. Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 1, Elementary Functions, Gordon & Breach, New York, 1986. Zhizhiashvili, L.V., Trigonometric Fourier Series and Their Conjugates, Kluwer Academic, Dordrecht, 1996. Zygmund, A., Trigonometric Series, 3rd Edition, Cambridge University Press, Cambridge, 2003. . Criteria of Uniform and Mean-Square Convergence of Fourier Series 8.4.3-1. Criteria of uniform convergence of Fourier series. LIPSCHITZ CRITERION. The Fourier series of a function f(x) converges uniformly. York, 1990. Mangulis, V., Handbook of Series for Scientists and Engineers, Academic Press, New York, 1965. Pinkus, A. and Zafrany, S., Fourier Series and Integral Transforms, Cambridge University. ∞. The motivation of using divergent asymptotic series of the form (8.5.1.2) is that a finite sum of the series (8.5.1.1) provides a good approximation for the given function for x →∞. For a fixed n,