8.3. POWER SERIES 353 A special case of the Taylor series (for x 0 = 0)istheMaclaurin series: ∞ n=0 1 n! f (n) (0)x n = f(0)+f (0)x + 1 2 f (0)x 2 + ··· . A formal Taylor series (Maclaurin series) for a function f (x)maybe: 1) divergent for x ≠ x 0 , 2) convergent in a neighborhood of x 0 to a function different from f (x), 3) convergent in a neighborhood of x 0 to the function f(x). In the last case, onesays that f(x) admits expansion in Taylor series in the said neighborhood, and one writes f(x)= ∞ n=0 1 n! f (n) (x 0 )(x – x 0 ) n . 8.3.2-2. Conditions of expansion in Taylor series. A necessary and sufficient condition for a function f(x) to be represented by its Taylor series in a neighborhood of a point x 0 is that the remainder term in the Taylor formula* should tend to zero as n →∞in this neighborhood of x 0 . In order that f(x) could be represented by its Taylor series in a neighborhood of x 0 , it suffices that all its derivatives in that neighborhood be bounded by the same constant, |f (n) (x)| ≤ M for all n. Uniqueness of the Taylor series expansion. For a function f (x) that can be represented as the sum of a power series, the coefficients of this series are determined uniquely (since this series is the Taylor series of f(x) and its coefficients have the form f (n) (x 0 ) n! ,where n = 0, 1, 2, ). Therefore, in problems of representing a function by a power series, the answer does not depend on the method adopted for this purpose. 8.3.2-3. Representation of some functions by the Maclaurin series. The following representations of elementary functions by Maclaurin series are often used in applications: e x = 1 + x + x 2 2! + x 3 3! + ···+ x n n! + ··· ; sin x = x – x 3 3! + x 5 5! – ···+(–1) n–1 x 2n–1 (2n – 1)! + ··· ; cos x = 1 – x 2 2! + x 4 4! – ···+(–1) n x 2n (2n)! + ··· ; sinh x = x + x 3 3! + x 5 5! + ···+ x 2n–1 (2n – 1)! + ··· ; * Different representations of the remainder in the Taylor formula are given in Paragraph 6.2.4-4. 354 SERIES cosh x = 1 + x 2 2! + x 4 4! + ···+ x 2n (2n)! + ··· ; (1 + x) α = 1 + αx + α(α – 1) 2! x 2 + ···+ α(α – 1) (α – n + 1) n! x n + ··· ; ln(1 + x)=x – x 2 2 + x 3 3 – ···+(–1) n+1 x n n + ··· ; arctan x = x – x 3 3 + x 5 5 – ···+(–1) n+1 x 2n–1 2n – 1 + ··· . The first five series are convergent for –∞ < x < ∞ (R = ∞), and the other series have unit radius of convergence, R = 1. 8.3.3. Operations with Power Series. Summation Formulas for Power Series 8.3.3-1. Addition, subtraction, multiplication, and division of power series. 1. Addition and subtraction of power series.Twoseries ∞ n=0 a n x n and ∞ n=0 b n x n with convergence radii R a and R b , respectively, admit term-by-term addition and subtraction on the intersection of their convergence intervals: ∞ n=0 a n x n ∞ n=0 b n x n = ∞ n=0 c n x n , c n = a n b n . The radius of convergence of the resulting series satisfies the inequality R c ≥ min[R a , R b ]. 2. Multiplication of power series.Twoseries ∞ n=0 a n x n and ∞ n=0 b n x n , with the respective convergence radii R a and R b , can be multiplied on the intersection of their convergence intervals, and their product has the form ∞ 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 . The convergence radius of the product satisfies the inequality R c ≥ min[R a , R b ]. 3. Division of power series. The ratio of two power series ∞ n=0 a n x n and ∞ n=0 b n x n , b 0 ≠ 0, with convergence radii R a and R b can be represented as a power series ∞ n=0 a n x n ∞ n=0 b n x n = c 0 + c 1 x + c 2 x 2 + ···= ∞ n=0 c n x n ,(8.3.3.1) whose coefficients can be found, by the method of indefinite coefficients, from the relation (a 0 + a 1 x + a 2 x 2 + ···)=(b 0 + b 1 x + b 2 x 2 + ···)(c 0 + c 1 x + c 2 x 2 + ···). 8.3. POWER SERIES 355 Thus, for the unknown c n , we obtain a triangular system of linear algebraic equations a n = n k=0 b k c n–k , n = 0, 1, , which is solved consecutively, starting from the first equation: c 0 = a 0 b 0 , c 1 = a 1 b 0 – a 0 b 1 b 2 0 , c n = a n b 0 – 1 b 0 n k=1 b k c n–k , n = 2, 3, The convergence radius of the series (8.3.3.1) is determined by the formula R 1 =min R a , ρ M + 1 , where ρ is any constant such that 0 < ρ < R b ; ρ can be chosen arbitrarily close to R b ;andM is the least upper bound of the quantities |b m /b 0 |ρ m (m = 1, 2, ), so that |b m /b 0 |ρ m ≤ M for all m. 8.3.3-2. Composition of functions representable by power series. Consider a power series z = f (y)=a 0 + a 1 y + a 2 y 2 + ···= ∞ n=0 a n y n (8.3.3.2) with convergence radius R. Let the variable y be a function of x that can be represented by a power series y = ϕ(x)=b 0 + b 1 x + a 2 x 2 + ···= ∞ n=0 b n x n (8.3.3.3) with convergence radius r. It is required to represent z as a power series of x and find the convergence radius of this series. Formal substitution of (8.3.3.3) into (8.3.3.2) yields z = f ϕ(x) = ∞ n=0 a n ∞ k=0 b k x k n = A 0 + A 1 x + A 2 x 2 + ···= ∞ n=0 A n x n ,(8.3.3.4) where A 0 = a 0 + a 1 b 0 + a 2 b 2 0 + ··· , A 1 = a 1 b 1 + 2a 2 b 0 b 1 + 3a 3 b 2 0 b 1 + ··· , A 2 = a 1 b 2 + a 2 (b 2 1 + 2b 0 b 2 )+3a 3 (b 0 b 2 1 + b 2 0 b 2 )+···, . 356 SERIES THEOREM ON CONVERGENCE OF SERIES (8.3.3.4). (i) If series (8.3.3.2) is convergent for all y (i.e., R = ∞ ), then the convergence radius of series (8.3.3.4) coincides with the convergence radius r of series (8.3.3.3). (ii) If 0 ≤ |b 0 | < R , then series (8.3.3.4) is convergent on the interval (–R 1 , R 1 ) ,where R 1 = (R – |b 0 |)ρ M + R – |b 0 | , and ρ is an arbitrary constant such that 0 < ρ < r ; ρ can be chosen arbitrarily close to r ;and M is the least upper bound of the quantities |b m |ρ m ( m = 1, 2, ), so that |b m |ρ m ≤ M for all m . (iii) If |b 0 | > R , then series (8.3.3.4) is divergent. Remark. Case (i) is realized, for instance if (8.3.3.2) has finitely many terms. 8.3.3-3. Local inversion of a function represented by power series. 1. Suppose that y = y(x) is a function that can be represented, in a neighborhood of a point x = x 0 , by the power series y = y 0 + a(x – x 0 )+b(x – x 0 ) 2 + c(x – x 0 ) 3 + d(x – x 0 ) 4 + ··· , a ≠ 0. Then the inverse function x = x(y), in a neighborhood of y = y 0 , can be represented by the series x = x 0 + 1 a (y – y 0 )– b a 3 (y – y 0 ) 2 + 2b 2 – ac a 5 (y – y 0 ) 3 + 5abc – 5b 3 – a 2 d a 7 (y – y 0 ) 4 + ··· . 2. B ¨ urman–Lagrange formula. Suppose that for a given function y = f (x), (8.3.3.5) the auxiliary function ϕ(x)= x f(x) is holomorphic in a neighborhood of the point x = 0 (i.e., it can be represented by a convergent power series in a neighborhood of that point). Then there is ε > 0 such that on the interval |y| < ε, the function (8.3.3.5) is invertible and its inverse x = g(y) is holomorphic on that interval, x = ∞ n=1 b n y n , b n = 1 n! d n–1 dx n–1 ϕ n (x) x=0 . The expression for the coefficients b n is called the B ¨ urman–Lagrange formula. Example 1. Consider the function y = x(x + b)(b ≠ 0), for which the auxiliary function has the form ϕ(x)=(x + a) –1 .UsingtheB ¨ urman–Lagrange formula and the relation d n–1 dx n–1 1 (x + a) n = (–1) n–1 n(n + 1) ···(2n – 2) (x + a) 2n–1 , we find the representation of the given function by power series: x = y a – y 2 a 3 + ···+(–1) n–1 (2n – 2)! (n – 1)! n! y n a 2n–1 + ··· . 8.4. FOURIER SERIES 357 8.3.3-4. Simplest summation formulas for power series. Suppose that the sum of a power series is known, ∞ k=0 a k x k = S(x). (8.3.3.6) Then, using term-by-term integration (on the convergence interval), one can fi nd the fol- lowing sums: ∞ k=0 a k k m x k = x d dx m S(x); ∞ k=0 a k (nk + m)x nk+m–1 = d dx x m S(x n ) ; ∞ k=0 a k nk + m x nk+m = x 0 x m–1 S(x n ) dx, n > 0, m > 0; ∞ k=0 a k nk + s nk + m x nk+s = x d dx x s–m x 0 x m–1 S(x n ) dx , n > 0, m > 0; ∞ k=0 a k nk + m nk + s x nk+s = x 0 x s–m d dx x m S(x n ) dx, n > 0, s > 0. (8.3.3.7) Example 2. Let us find the sum of the series ∞ n=0 kx k–1 . We start with the well-known formula for the sum of an infinite geometrical progression: ∞ k=0 x k = 1 1 – x (|x| < 1). This series is a special case of (8.3.3.6) with a k = 1, S(x)=1/(1 – x). The series ∞ n=0 kx k–1 can be obtained from the left-hand side of the second formula in (8.3.3.7) for m = 0 and n = 1. Substituting S(x)=1/(1 – x) into the right-hand side of that formula, we get ∞ k=0 kx k–1 = d dx 1 1 – x = 1 (1 – x) 2 (|x| < 1). 8.4. Fourier Series 8.4.1. Representation of 2π-Periodic Functions by Fourier Series. Main Results 8.4.1-1. Dirichlet theorem on representation of a function by Fourier series. A function f (x) is said to satisfy the Dirichlet conditions on an interval (a, b)if: 1) this interval can be divided into finitely many intervals on which f (x) is monotone and continuous; 2) at any discontinuity point x 0 of the function, there exist finite one-sided limits f(x 0 + 0)andf(x 0 – 0). 358 SERIES DIRICHLET THEOREM. Any 2π -periodic function that satisfies the Dirichlet conditions on the interval (–π, π) can be represented by its Fourier series f(x)= a 0 2 + ∞ n=1 a n cos nx + b n sin nx (8.4.1.1) whose coefficients are defined by the Euler–Fourier formulas a n = 1 π π –π f(x)cosnx dx, n = 0, 1, 2, , b n = 1 π π –π f(x)sinnx dx, n = 1, 2, 3, (8.4.1.2) At the points of continuity of f(x) , the Fourier series converges to f(x) , and at any discontinuity point x 0 , the series converges to 1 2 [f(x 0 + 0)+f (x 0 – 0)] . The coefficients a n and b n of the series (8.4.1.1) are called the Fourier coefficients. Remark. Instead of the integration limits –π and π in (8.4.1.2), one can take c and c + 2π,wherec is an arbitrary constant. 8.4.1-2. Lipschitz and Dirichlet–Jordan convergence criteria for Fourier series. LIPSCHITZ CRITERION. Suppose that f(x) is continuous at a point x 0 and for sufficiently small ε > 0 satisfies the inequality |f(x 0 ε)–f(x 0 )| ≤ Kε σ ,where L and σ are constants, 0 < σ ≤ 1 . Then the representation (8.4.1.1)–(8.4.1.2) holds at x = x 0 . In particular, the conditions of the Lipschitz criterion hold for continuous piecewise differentiable functions. Remark. The Fourier series of a continuous periodic function with no additional conditions (for instance, of its regularity) may happen to be divergent at infinitely many (even uncountably many) points. DIRICHLET–JORDAN CRITERION. Suppose that f(x) is a function of bounded variation on some interval (x 0 – h, x 0 + h) (–π, π) (i.e., f(x) can be represented as a difference of two monotonically increasing functions). Then the Fourier series (8.4.1.1)–(8.4.1.2) of the function f(x) at the point x 0 converges to the value 1 2 [f(x 0 + 0)+f (x 0 – 0)] . 8.4.1-3. Riemann localization principle. RIEMANN LOCALIZATION PRINCIPLE. The behavior of the Fourier series of a function f(x) at a point x 0 * depends only on its values near that point, i.e., values in an arbitrarily small neighborhood of that point. Thus, for two functions that coincide in a neighborhood of a point x 0 , but differ outside that neighborhood, the corresponding Fourier series at x 0 are either both convergent or divergent and have the same sum in the case of convergence, although their Fourier coefficients may be different, being dependent on all values of the functions. * What is meant here is the fact of convergence or divergence of the Fourier series at x 0 ,andalsothe numerical value of its sum in the case of convergence. 8.4. FOURIER SERIES 359 8.4.1-4. Asymptotic properties of Fourier coefficients. 1 ◦ . Fourier coefficients of an absolutely integrable function tend to zero as n goes to infinity: a n → 0 and b n → 0 as n →∞. 2 ◦ . Fourier coefficients of a continuous 2π-periodic function have the following limit properties: lim n→∞ (na n )=0, lim n→∞ (nb n )=0, i.e., a n = o(1/n), b n = o(1/n). 3 ◦ . If a continuous periodic function is continuously differentiable up to the order m – 1 inclusively, then its Fourier coefficients have the following limit properties: lim n→∞ (n m a n )=0, lim n→∞ (n m b n )=0, i.e., a n = o n –m , b n = o n –m . 8.4.1-5. Integration and differentiation of Fourier series. 1 ◦ . The Fourier series of a continuous periodic function of bounded variation admits term- by-term integration, and the resulting series is uniformly convergent. 2 ◦ . The Fourier series of a k times continuously differentiable function admits term-by- term differentiation (k – 1) times, the resulting series still being uniformly convergent (the kth differentiation yields the kth derivative of the function, but the resulting series may have only mean-square convergence, not necessarily pointwise convergence). 8.4.2. Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions 8.4.2-1. Expansion of 2l-periodic and nonperiodic functions in Fourier series. 1 ◦ . The case of 2l-periodic functions can be easily reduced to that of 2π-periodic functions by changing the variable x to z = πx l . In this way, all the results described above for 2π-periodic functions can be easily extended to 2l-periodic functions. The Fourier expansion of a 2l-periodic function f(x)hastheform f(x)= a 0 2 + ∞ n=1 a n cos nπx l + b n sin nπx l ,(8.4.2.1) where a n = 1 l l –l f(x)cos nπx l dx, b n = 1 l l –l f(x)sin nπx l dx.(8.4.2.2) 2 ◦ . A nonperiodic (aperiodic) function f(x)defined on the interval (–l, l) can also be represented by a Fourier series (8.4.2.1)–(8.4.2.2); however, outside that interval, the sum of that series S(x) may differ from f (x)*. *ThesumS(x)isa2l-periodic function defined for all x,butf(x) may happen to be nonperiodic, or even undefined outside the interval (–l, l). . series ∞ n=0 kx k–1 can be obtained from the left-hand side of the second formula in (8.3.3.7) for m = 0 and n = 1. Substituting S(x)=1/(1 – x) into the right-hand side of that formula, we get ∞ k=0 kx k–1 = d dx 1 1. subtraction, multiplication, and division of power series. 1. Addition and subtraction of power series.Twoseries ∞ n=0 a n x n and ∞ n=0 b n x n with convergence radii R a and R b , respectively,. dependent on all values of the functions. * What is meant here is the fact of convergence or divergence of the Fourier series at x 0 ,andalsothe numerical value of its sum in the case of convergence. 8.4.