Laurent and Fourier Series

3.2.1 Laurent and Fourier Series

A Laurent series is a series of the form

n =−∞

Its convergence region is the intersection of the convergence regions of the expansions

the interior of which are determined by conditions of the form |z| < r 2 and |z| > r 1 . The

convergence region can be void, for example, if r 2 <r 1 .

If 0 < r 1 <r 2 < ∞, then the convergence region is an annulus, r 1 < |z| < r 2 . The series defines an analytic function in the annulus. Conversely, if f (z) is a single-valued analytic function in this annulus, it is represented by a Laurent series, which converges uniformly in every closed subdomain of the annulus.

The coefficients are determined by the following formula, due to Cauchy: 55

c n = z −n−1 f (z)dz, r 1 <r<r 2 , −∞ < n < ∞ (3.2.2) 2πi |z|=r

and |c n |≤r −n max |f (z)|.

|z|=r

The extension to the case when r 2 = ∞ is obvious; the extension to r 1 = 0 depends on whether there are any terms with negative exponents or not. In the extension of formal power series to formal Laurent series, however, only a finite number of terms with negative indices are allowed to be different from zero; see Henrici [196, Sec. 1.8]. If you substitute z for z −1 an infinite number of negative indices is allowed, if the number of positive indices is finite.

Example 3.2.1.

A function may have several Laurent expansions (with different regions of conver- gence), for example,

if |z| < |a|, (z − a) −1

=0 a −n−1 z

if |z| > |a|. The function 1/(z − 1) + 1/(z − 2) has three Laurent expansions, with validity conditions

m =1 a m −1 z −m

|z| < 1, 1 < |z| < 2, 2 < |z|, respectively. The series contains both positive and negative powers of z in the middle case only. The details are left for Problem 3.2.4 (a).

55 Augustin Cauchy (1789–1857) is the father of modern analysis. He is the creator of complex analysis, in which this formula plays a fundamental role.

192 Chapter 3. Series, Operators, and Continued Fractions

Remark 3.2.1.

The restriction to single-valued analytic functions is important in this subsection. In this book we cannot entirely avoid working with multivalued functions such as √z, ln z, z α , (α noninteger). We always work with such a function, however, in some region where one branch of it, determined by some convention, is single-valued. In the examples mentioned, the natural conventions are to require the function to be positive when z>

1, and to forbid z to cross the negative real axis. In other words, the complex plane has a cut along the negative real axis. The annulus mentioned above is incomplete in these cases; its intersection with the negative real axis is missing, and we cannot use a Laurent expansion.

For a function like ln( z +1 z −1 ) , we can, depending on the context, cut out either the interval [−1, 1] or the complement of this interval with respect to the real axis. We then

use an expansion into negative or into positive powers of z, respectively. If r 1 < 1<r 2 , we set F (t) = f (e it ) . Note that F (t) is a periodic function;

F (t + 2π) = F (t). By (3.2.1) and (3.2.2), the Laurent series then becomes for z = e it a Fourier series :

F (t ) =

c n e int , c n =

e −int

F (t ) dt. (3.2.4)

n =−∞

Note that c −m m = O(r 1 ) for m → +∞, and c n = O(r −n

2 ) for n → +∞. The formulas in (3.2.4), however, are valid in much more general situations, where c n → 0 much more slowly, and where F (t) cannot be continued to an analytic function f (z), z = re it , in an

annulus. (Typically, in such a case r 1 =1=r 2 .)

A Fourier series is often written in the following form:

F (t ) = a 0 +

(a k cos kt + b k sin kt). (3.2.5)

2 k =1

Consider c k e ikt +c −k e −ikt ≡a k cos kt +b k sin kt. Since e ±ikt = cos kt ±i sin kt, we obtain for k ≥ 0

π −π (3.2.6) Also note that a k − ib k = 2c k . If F (t) is real for t ∈ R, then c −k = ¯c k . We mention without proof the important Riemann–Lebesgue theorem,

56, 57 by which the Fourier coefficients c n tend to zero as n → ∞ for any function that is integrable (in the

sense of Lebesgue), a fortiori for any periodic function that is continuous everywhere. A finite number of finite jumps in each period are also allowed.

A function F (t) is said to be of bounded variation in an interval if, in this interval, it can be expressed in the form F (t) = F 1 (t ) −F 2 (t ) , where F 1 and F 2 are nondecreasing

56 George Friedrich Bernhard Riemann (1826–1866), a German mathematician, made fundamental contributions to analysis and geometry. In his habilitation lecture 1854 in Göttingen, Riemann introduced the curvature tensor

and laid the groundwork for Einstein’s general theory of relativity. 57 Henri Léon Lebesgue (1875–1941), a French mathematician, created path-breaking general concepts of mea-

sure and integral.

3.2. More about Series 193 bounded functions. A finite number of jump discontinuities are allowed. The variation of

F b over the interval [a, b] is denoted

|dF (t)|. If F is differentiable the variation of F equals a |F ′ (t ) | dt.

Another classical result in the theory of Fourier series reads as follows: If F (t) is of bounded variation in the closed interval [−π, π], then c n = O(n −1 ) ; see Titchmarsh [351,

Secs. 13.21, 13.73]. This result can be generalized as the following theorem.

Theorem 3.2.1.

Suppose that F (p) is of bounded variation on [−π, π], and that F (j ) is continuous everywhere for j < p. Denote the Fourier coefficients of F (p) (p) (t ) by c

n . Then

(3.2.7) Proof. The theorem follows from the above classical result, after the integration of the

c n (p) = (in) −p c n = O(n −p−1 ).

formula for c n in (3.2.2) by parts p times. Bounds for the truncation error of a Fourier series can also be obtained from this. The

details are left for Problem 3.2.4 (d), together with a further generalization. A similar result is that c n = o(n −p ) if F (p)

is integrable, hence a fortiori if F ∈ C p . In particular, we find for p = 1 (since

n −2 is convergent) that the Fourier series (3.2.2) converges absolutely and uniformly in R. It can also be shown that the Fourier series is valid , i.e., the sum is equal to F (t).