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Unit 6 Classification of Singularities and Calculus of Residues

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Unit 6 Classification of Singularities and Calculus of Residues UNIT 6 CLASSIFICATION OF SINGULARITIES AND CALCULUS OF RESIDUES Structure 6.1 Introduction Objectives 6.2 Classification of Singularities 6.2.1 Zeros of an Analytic Function 6.2.2 Singular P~ints 6.2.3 Types of Isolated Singular Points 6.2.4 Singularities at Infinity 6.3 Calculus of Residues 6.3.1 Residue at a Finite Point 6.3.2 Residue at the Point at Infmity 6.3.3 Cauchy 's Residue Theorem 6.3.4 Mittag-I,eflk-rExpansion Theorem 6.3.5 Rouche's Theorem 6.4 Summary 6.5 Answers/Solutions/Hints to Exercises 6.1 INTRODUCTION Consider the functions 1 1 1 ;,x sin ;, exp (- k2), ---- etc. x ER x(x - l)(x - 2)' We see that each of these fmctions is not defrned at 0. In such a case, the point x = 0 is a singular point for each of these functions in the sense that the function is defined in a deleted neighbourhood of 0. The problem of classifying singularities are not easy to solve satisfactorily for functions on R. On the other hand, the situation in C is different. Iff is analylrc at a point s = a, then there is a neighbourhood N of a inside which f is analytic. Let r be a positively oriented smooth closed curve contained in N. Then, f is analybc Inside and on r and, as seen in Unit 5, celebrated Cauchy Integral Theorem tells us that If, however,ffails to be analytic at finitely many points interior to r, then the above argument fails; which means that there is, as we shall see in this unit, a specified number (zero or non-zero) which each of these points contribute to the value of the integral. This motivates to generalize the idea of Cauchy Integral Theorem to functions which have isolated singularities. This is reflected in the Residue Theorem. In this unit, we start with the zeros of an analyhc function. We then classify the singularities of complex-valued functions - in particular, we define isolated and non-isolated singuldies, removable singularities, poles and their order, singulsrrities at infinity and illustrate these singularities. Cm~plbxVariables The notion of residue at an isolated singularity is introduced and this notion is used to 1 prove the Residue Theorem. Using theresidue theorem, we develop and illustrate some of the basic methods employed in Complex Integration. Objectives b 1 After studying this unit, you should be able to I I tell zeros of an analytic function and find their order, define sinnularities of an analytic function and make distinction between (i) isolated and non- isolated singularities (ii) removable and irremovable singularities, -1 determine poles, if any, of an analytic hction and their order, define singularities at infinity, define and obtain residue of an analytic function at a frnite point, define and obtain residue at the point at infinity, learn Residue Theorem and its applications in evaluating some integrals in complex plane, and learncertain applications of residue theorem. 6.2 CLASSIFICATION OF SINGULARITIES Before we take up singularities of a complex-valued function, it is desirable to study. the zeros of an analytic function, which we take up next. 6.2.1 Zeros of an Analytic Function If a function f(z), analytic in a region R, is zero at a point zo in R, then zo is called a zero of f(z). If f(zo) = 0 but f '(z,) # 0, then z, is called a simple zero or a zero of the first order. If f(zo) = 0, f '(z0) = 0 ,..., f (n- ') (ZO)= 0, but f ("' (z,) + 0, then z, is called a zero of order n. For ixample, the function z2 sin z has a zero of order three at z = 0 and simple zeros at The function 1 - cos z has second order zeros at z = 0, + 2n, +- 4n, .. ' If an analytic function f(z) has a zero of order n at z = z,, then its Taylor series is of the form 2 =(~-~O)~[b,+h,-l(z-~~)+b,+~(z-z~) +...I = (z - z,)" [b, + (Z - zO)Sj, say By taking ) z - z, ) sufficiently small, we can make lb,dl > lz - zol . I SI Thus a neighbourhood of zo can be found where is not zero. Consequently, f (zl kasnrrcrt have another zero in this neighbourhood. ClvssiCication of Singularities and Calculus uf Residues We express this by saying that the zeros of an analytic function are isolated. We next take singularities of complex-valued functions 6.2.2 Singular Points A complex-valued function f(z) is said to have a regular point at z, iff is analytic at z, . The point zo is called a Singular Point or Singularity of f(z) if z, is a limit point of regular points and f(z) is not analytic at zo . For example z = 0 is a singular point off(z) = z-', z # 0. If the singular point z, is such. that there is no other singular point in its neighbourhood, i.e., iff (z) is analytic in some deleted neighbourhood of zo ,then the point z, is called an Isolated Singular Point or Isslated Singularity off (2). For example, the functionflz) = lis analytic eveqwhere except at z = a.Thus f(z) z - a has an isolated singularity at z = a. z2 + 1 T!le function has three isolated singular points, namely (z- 1) (z +4) The function -has an infinite number of isolated singularities at sin(%) z = f 1, + V2, k V3, f l/4, ....... The origin z = 0 is also a singular point, but it is not an isolated singular point, because there are other singular points in its neighbourhood, however small we choose thf;neighbourhood. The function &has a singularity at z = 0, which is not isolated. The other si~lgularities near the origin arise due to the discontinuity suffered by a single-valued branch of & on crossing the cut in the z-plane. Singularities of this nature always arise at 'branch points', where two branches of a multivalued function have the same value. Other examples of such a singularity are log z'at z = 0 and sin-'z at z = f 1. If ro is not an isolated singularity off (z), it is called a Non-isolated singular point of f(z). Descriptions of Non-isolated singularity at z, and for isolated singularities at z,,I,, 2, are given in Figures 6.1 & 6.2, respectively. Example 6.1 "Discuss the singularities if'g(z) = '/f(z), where Az) = U(X,y) + iv(x, y) = sin Solution It is easy to see thatf (z) is analybc everywhere except at z = 0. Therefore the singularity of g is at z = 0 and at the points where f (z) = 0, i.e., when u (x,y) = 0 and v(x, y) = 0. Since cos h @/I a f ) > 1, therefore u (x, y) = 0 implies that In the case when sin(x/( z 12) = 0,we have cos( ?flzl2 ) f 0. Thus v(x,y) = 0 implies that sinh (v/lz12)= 0 Thus (6.1) gives and (6.2) yields y = 0. I+ I I 1 Further,y=Ogivesx=.-forn=O, + 1,-+2,........ Thusu(x,y)=Oand (n X) v(x, y)=Oholdsifandonly if x=l/(nn),n=O ,+1 ,+2 ,... .Thusthe singularities ofg(z) are at points x = l/(nn) , n = 0,+ 1 ,f 2 ,... and at their limit point z = 0. This shows that g (z) has isolated singularities at z = l/nn, n = 0, f 1, f 2, .... and has non-isolated singularity at the limit point z = 0. Remark 1 :We note that fi)= sin the discussion of singularities of g zeros of sin ( l/z ). Remark 2 : Any neighbourhood of a non-isolated singular point of a function f (2) contains other singularities and hence a non-isolated singular point off (2) is a limit point of the singular points of f(z). Further observe that f(z) must be discontinuous at an isolated singular point. You may now try an exercise. Exercise 1 Discuss tlte singulnritiet;PC 1/COS ( ). We next take up the kinds of isolated singularities for an analytic function. 6.2.3 Types of Isolated Singular Points There are three kinds of isolated singularities for an analytic function i) Removable ,Singttlurity, \vhicli upon close exalniilatioii is not actually a singular point at all. ii) Pole is a zero of the reciprocal which is an analytic iimctiun. iii) Erseitial sing?tlnrit.v, which is neither rerl~ovablenor a pole. We next,give definition and examples of these kinds of isolated singularities. Definition : "An isolated singularity zz,of C (2). f E N ( D \, I x-, j) , is calied rcniovable or that f (z) has a removable singularity at z, iff (z) can be defr12ed at z, so that it becomes analytic at z, '' In other words, f (2) can be extended to z, too so that f (2) is analy~icon D Xote : :V ( D\lx, j) is the dclctcd neighbourhood, 1.e D escluding the po~nt2, Consider the fui~ctious. .I;(z ) = sin z, 2 # 1 Thesc ftlncdnns 11a-c.cremovable singularitics at 1. 1 and O respcc1ive1.c.w!~icl~ call bc rcrllovcrt by letting ,/;(I) =sinl, J: (1) := 2 and I; (0):~1, ccs))ec~i\~cly Exercise 2 " Discuss the removable singu1,uities sf the functions sin z L 2 (1 -cos z) and g, (zj = -- zf0 z2 and give the values of the functions ~v11ichcan remove the singularities ". We obscrvc that if J{L)is analyl~con ol)cn scl S and z,, E S .
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