Applied Mathematics E-Notes, 13(2013), 36-50 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ On A Generalization Of The Laurent Expansion Theorem Branko Sari´cy Received 19 March 2012 Abstract Based on the properties of the non-univalent conformal mapping ez = s a causal connection has been established between the Laurent expansion theorem and the Fourier trigonometric series expansion of functions. This connection combined with two highly significant results proved in the form of lemmas is a foundation stone of the theory. The main result is in the form of a theorem that is a natural generalization of the Laurent expansion theorem. The paper ends with a few examples that illustrate the theory. 1 Introduction A trigonometric series is a series of the form + a 1 0 + [a cos (k) + b sin (k)] (1) 2 k k k=1 X where the real coeffi cients a0, a1, ..., b1, b2, ... are independent of the real variable . Applying Euler’s formulae to cos (k) and sin (k), we may write (1) in the complex form + 1 ik cke (2) k= X1 where 2ck = ak ibk and for k ( is the set of natural numbers) and 2c0 = a0. 2 @ @ Here c k is conjugate to ck [8]. Letz = +i be a complex variable. Suppose that series (2) converges at all points of the interval ( , ) to a real valued point function f (). If f ( iz) is integrable on the interval = z = 0, [ , ] , then the Fourier coeffi cients ck of f () are determined= uniquely f asj 2 g 1 kz ck = f ( iz) e dz (k = 0, 1, 2, ...). (3) 2i ± ± Z= Mathematics Subject Classifications: 15A15, 42A168,; 30B10, 30B50. yFaculty of Sciences, University of Novi Sad. Trg Dositeja Obradovica 2, 21000 Novi Sad, Serbia; College of Technical Engineering Professional Studies, Svetog Save 65, 32 000 Caµ µcak, Serbia 36 B. Sari´c 37 + z k Similarly, if for = 0 a trigonometric series k=1 c^k (e ) converges in ( , ) to a z 1 complex valued point function f (e ), which is integrable on , then P = 1 z kz c^k = f (e ) e dz (k = 0, 1, 2, ...). (4) 2i ± ± Z= As is well-known, progress in Fourier analysis has gone hand in hand with progress in theories of integration. Perhaps this can be best exemplified by using the so-called total value of the generalized Riemann integrals [5, 6]. This brand new theory of integration, which takes the notion of residues of real valued functions into account, gives us the opportunity to integrate real valued functions that was not integrable in any of the known integration methods until now. All of this leads to the following very important problem. Let be a simple closed contour in the z-plane, which consists of C and any regular curve Q connecting the endpoints of , inside and on which a= complex valued point function f (z) is analytic except for a= finite number of isolated singular points z1, z2, ..., zn belonging to the interior (int) of . Chose > 0 to be + = small enough so that Q and the semi-circumferences = z z z = ρ, > 0 f j j j g and = z z z = ρ, < 0 ( = 1, 2, ..., n) are disjoint. Having formed the f j j j g numbers c^k by means of 1 kz c^k = vt f (z) e dz 2i Z= n n y kz 1 f (z) e dz kz = lim f (z) e dz + x kz (5) 2i 0+ + f (z) e dz ! "=0 l =1 ( R # X Z X R for k = 0, 1, 2, ... where vt denotes the total value of an improper integral, l0 = ± ± [ i, z1 i], l = [z + iρ, z+1 i] ( = 1, 2, ..., n 1) and ln = [zn + iρ, i], we + ik may call the series k=1 c^ke the Fourier series of f (z) on . This means that 1 = the numbers c^k are connected with f (z) by the formula (5). Clearly, in this case, the P numbers c^k, which are the Fourier coeffi cients of f (z), are not determined uniquely since on account of Cauchy’s residue theorem y kz kz 0 vt f (z) e dz = f (z) e dz + n kz (6) 2i Resz=z [f (z) e ] Q =1 Z= Z P whenever Q is in the left half plane of the z-plane. A question that arises is whether this Fourier series of f (z) does in fact converge and if it does not whether it is summable in any other sense and under what conditions is its sum equal to f (z) in int z1, z2, ..., zn . As we shall see, in what follows, the answer to this question is closely= relatednf to the gLaurent expansion theorem. 2 Main Results By the Laurent expansion theorem, a single-valued complex function f (s) of a complex variable s, analytic in an arbitrary punctured disc s 0 < s < R of the s-plane, f j j j g 38 On a Generalization of the Laurent Expansion Theorem + k has a Laurent series expansion f (s) = k=1 c^ks . The Laurent coeffi cients c^k are determined uniquely as 1 P 1 f (s) c^k = k+1 ds (k = 0, 1, 2, ...) (7) 2i r s ± ± ZC where r = s : s = r, 0 < r < R [3]. On the other hand, it is well-known that the horizontalC f stripj j = z = + ig : ( , ) , [ , ] of the z-plane is mapped onto the puncturedD f disc ? = s :2 0 <1s < e 2in the sg-plane by the non- univalent conformal mapping s = Dez. Accordingly,f j j if a single-valuedg complex function f (s) is analytic in ? and > 0, then, on account of the Laurent expansion theorem, D+ z k the Fourier series k=1 c^k (e ) with the Fourier coeffi cients (4) converges in to f (ez). Furthermore, it1 follows from (7) that for any real number < D P +i 1 z kz c^k = f (e ) e dz (k = 0, 1, 2, ...). (8) 2i i ± ± Z Here a single-valued complex function f (s) is taken to be the function f (ez) of period 2 with respect to in the z-plane. Generally, a complex valued point function f (z) is said to be integrable, of bounded variation and the like, with respect to either or , in the strip , if its real and imaginary parts satisfy separately the aforementioned properties forD all fixed values of the second variable in . As is well known, theD first theorem, chronologically, in the theory of Fourier trigono- metric series, is as follows [8]. THEOREM 1. If a real valued point function f is of bounded variation, then the Fourier series of f converges at every point to the value 1 lim [f ( + ) + f ( )] . (9) 2 0+ ! If f is in addition continuous at every point on an interval I = [a, b], then the Fourier series of f is uniformly convergent in I. The result of the preceding theorem implies the following result. THEOREM 2. If a single valued complex function f (s) is of bounded variation with respect to arg s in a punctured disc = s : 0 < s < e of the s-plane, then + k D f j j g the Laurent series k=1 c^ks , with the coeffi cients 1 P 1 f (s) c^k = k+1 ds (k = 0, 1, 2, ...), (10) 2i s ± ± ZC where = s : s = e , < , converges at every point s to the value C f j j g 2 C 1 i i lim f se + f se . (11) 2 0+ ! B. Sari´c 39 If f is in addition continuous on , then the Laurent series of f is uniformly convergent C on . C This result strongly suggests that single valued complex functions having isolated singularities inside may be expanded in the Fourier trigonometric series, too. Let’sshow this works.= Let w be a complex parameter independent of z = + i. As is well known, if a real valued point function f () is of bounded variation in the interval [ , ], then for ^ ( , ) and Re(w) 0 there exist finite limits, [3], 2 ≥ iw(i^ z) lim w e f ( iz) dz = i lim f(^ ) w + ^ 0+ j j! 1 Z= ! and iw(i^ z) lim w e f ( iz) dz = i lim f(^ + ) (12) w + 0+ j j! 1 Z=^ ! ^ ^ ^ where = z = + i : = 0, [ , ] and ^ = z = + i : = 0, [, ] . = f 2 g = f 2 g Since the above conditions play a key role in the proof of Theorem 1 based on the Cauchy calculus of residues, see [3], it follows that all functions that satisfy these conditions have a Fourier series expansion + f ( + ) + f ( ) 1 ik lim = c^ke , 0+ 2 ! k= X1 with the Fourier coeffi cients (4). Furthermore, pursuing the matter in the proof of Theorem 1, one easily comes to a conclusion that conditions (12) cannot be satisfied for Re(w) 0 but only for Re(w) > 0. This is an immediate consequence of the result presented in≥ the following form. LEMMA 1. For w C, let a single valued complex function f (w) be analytic in the w-plane with the2 exception of an infinite but countable number of isolated i singularities wk (k = 1, 2, ...) on the half-straight lines w = w e : w > 0, = ^ and f j j j j g w = w ei : w > 0, = +^ by which the w-plane is divided into two open adjacent f j j j j g regions = w = w ei : w > 0, + ^ < < ^ and = w = w ei : w > 0, ^ R ^f j j j j g L f j j j j < < + .