supplement - Some Applications of the Residue Theorem...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Some Applications of the Residue Theorem * Supplementary Lecture Notes MATH 322, Complex Analysis Winter 2005 Pawe± l Hitczenko Department of Mathematics Drexel University Philadelphia, PA 19104, U.S.A. email: phitczenko@math.drexel.edu * I would like to thank Frederick Akalin for pointing out a couple of typos. 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
1 Introduction These notes supplement a freely downloadable book Complex Analysis by George Cain (henceforth referred to as Cain’s notes), that I served as a primary text for an undergraduate level course in complex analysis. Throughout these notes I will make occasional references to results stated in these notes. The aim of my notes is to provide a few examples of applications of the residue theorem. The main goal is to illustrate how this theorem can be used to evaluate various types of integrals of real valued functions of real variable. Following Sec. 10.1 of Cain’s notes, let us recall that if C is a simple, closed contour and f is analytic within the region bounded by C except for finitely many points z 0 ,z 1 ,...,z k then Z C f ( z ) dz = 2 πi k X j =0 Res z = z j f ( z ) , where Res z = a f ( z ) is the residue of f at a . 2 Evaluation of Real-Valued Integrals. 2.1 Definite integrals involving trigonometric functions We begin by briefly discussing integrals of the form Z 2 π 0 F (sin at, cos bt ) dt. (1) Our method is easily adaptable for integrals over a different range, for example between 0 and π or between ± π . Given the form of an integrand in (1) one can reasonably hope that the integral results from the usual parameterization of the unit circle z = e it , 0 t 2 π . So, let’s try z = e it . Then (see Sec. 3.3 of Cain’s notes), cos bt = e ibt + e - ibt 2 = z b + 1 /z b 2 , sin at = e iat - e - iat 2 i = z a - 1 /z a 2 i . Moreover, dz = ie it dt , so that dt = dz iz . Putting all of this into (1) yields Z 2 π 0 F (sin at, cos bt ) dt = Z C F ± z a - 1 /z a 2 i , z b + 1 /z b 2 ² dz iz , where C is the unit circle. This integral is well within what contour integrals are about and we might be able to evaluate it with the aid of the residue theorem. 2
Background image of page 2
It is a good moment to look at an example. We will show that Z 2 π 0 cos 3 t 5 - 4 cos t dt = π 12 . (2) Following our program, upon making all these substitutions, the integral in (1) becomes Z C ( z 3 + 1 /z 3 ) / 2 5 - 4( z + 1 /z ) / 2 dz iz = 1 i Z C z 6 + 1 z 3 (10 z - 4 z 2 - 4) dz = - 1 2 i Z C z 6 + 1 z 3 (2 z 2 - 5 z + 2) dz = - 1 2 i Z C z 6 + 1 z 3 (2 z - 1)( z - 2) dz. The integrand has singularities at z 0 = 0, z 1 = 1 / 2, and z 2 = 2, but since the last one is outside the unit circle we only need to worry about the first two. Furthermore, it is clear that z 0 = 0 is a pole of order 3 and that z 1 = 1 / 2 is a simple pole. One way of seeing it, is to notice that within a small circle around z 0 = 0 (say with radius 1 / 4) the function z 6 + 1 (2 z - 1)( z - 2) is analytic and so its Laurent series will have all coefficients corresponding to the negative powers of z zero. Moreover, since its value at
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 14

supplement - Some Applications of the Residue Theorem...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online