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 BrankoSari·c y Received 19 March 2012 Abstract Based on the properties of the non-univalent conformal mapping e z = 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 signicant 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 0 2 + +1 X k=1 [a k cos (k )+ b k sin (k )] (1) where the real coe¢cients a 0 ;a 1 ; :::; b 1 ;b 2 ; ::: are independent of the real variable . Applying Eulers formulae to cos (k ) and sin (k ), we may write (1) in the complex form +1 X k=1 c k e ik (2) where 2c k = a k ib k and for k 2@ (@ is the set of natural numbers) and 2c 0 = a 0 . Here c k is conjugate to c k [8]. Let z = + 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 = = fz j =0; 2 [;]g, then the Fourier coe¢cients c k of f ( ) are determined uniquely as c k = 1 2i Z = f (iz) e kz dz (k =0; 1; 2; :::): (3) Mathematics Subject Classications: 15A15, 42A168,; 30B10, 30B50. y Faculty 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 Ca cak, Serbia 36