Chap 6 Residues and Poles if f analytic. Cauchy-Goursat Theorem: What if f is not analytic at finite number of points interior to C Residues. 53. Residues z0 is called a singular point of a function f if f fails to be analytic at z0 but is analytic at some point in every neighborhood of z0. A singular point z0 is said to be isolated if, in addition, there is a deleted neighborhood of z0 throughout which f is analytic. tch-prob

The origin is a singular point of Log z, but is not isolated Ex1. Ex2. The origin is a singular point of Log z, but is not isolated Ex3. not isolated isolated When z0 is an isolated singular point of a function f, there is a R2 such that f is analytic in tch-prob

Consequently, f(z) is represented by a Laurent series and C is positively oriented simple closed contour When n=1, The complex number b1, which is the coefficient of in expansion (1) , is called the residue of f at the isolated singular point z0. A powerful tool for evaluating certain integrals. tch-prob

Ex4. 湊出z-2在分母 tch-prob

Ex5. tch-prob

54. Residue Theorems Thm1. Let C be a positively oriented simple closed contour. If f is analytic inside and on C except for a finite number of (isolated) singular points zk inside C, then Cauchy’s residue theorem tch-prob

Ex1. tch-prob

Thm2: If a function f is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then Pf: tch-prob

Ex2. tch-prob

55. Three Types of Isolated Singular points If f has an isolated singular point z0, then f(z) can be represented by a Laurent series tch-prob

(i) Type 1. Ex1. tch-prob

is known as a removable singular point. Ex2. (ii) Type 2 bn=0, n=1, 2, 3,…… is known as a removable singular point. * Residue at a removable singular point is always zero. tch-prob

* If we redefine f at z0 so that f(z0)=a0 define Above expansion becomes valid throughout the entire disk * Since a power series always represents an analytic function Interior to its circle of convergence (sec. 49), f is analytic at z0 when it is assigned the value a0 there. The singularity at z0 is therefore removed. Ex3. tch-prob

Infinite number of bn is nonzero. (iii) Type 3: Infinite number of bn is nonzero. is said to be an essential singular point of f. In each neighborhood of an essential singular point, a function assumes every finite value, with one possible exception, an infinite number of times. ~ Picard’s theorem. tch-prob

has an essential singular point at where the residue Ex4. has an essential singular point at where the residue an infinite number of these points clearly lie in any given neighborhood of the origin. tch-prob

an infinite number of these points clearly lie in any given neighborhood of the origin. tch-prob

56. Residues at Poles identify poles and find its corresponding residues. Thm. An isolated singular point z0 of a function f is a pole of order m iff f(z) can be written as tch-prob

Pf: “<=“ tch-prob

“=>” tch-prob

Ex1. tch-prob

Need to write out the Laurent series for f(z) as in Ex 2. Sec. 55. tch-prob

Ex4. tch-prob

57. Zeros and Poles of order m Consider a function f that is analytic at a point z0. (From Sec. 40). Then f is said to have a zero of order m at z0. tch-prob

Thm. Functions p and q are analytic at z0, and Ex1. Thm. Functions p and q are analytic at z0, and If q has a zero of order m at z0, then has a pole of order m there. tch-prob

Corollary: Let two functions p and q be analytic at a point z0. Ex2. Corollary: Let two functions p and q be analytic at a point z0. Pf: Form Theorem in sec 56, tch-prob

The singularities of f(z) occur at zeros of q, or Ex3. The singularities of f(z) occur at zeros of q, or try tan z tch-prob

Ex4 tch-prob

58. Conditions under which Lemma : If f(z)=0 at each point z of a domain or arc containing a point z0, then in any neighborhood N0 of z0 throughout which f is analytic. That is, f(z)=0 at each point z in N0. Pf: Under the stated condition, For some neighborhood N of z0 f(z)=0 Otherwise from (Ex13, sec. 57) There would be a deleted neighborhood of z0 throughout which tch-prob

Since in N, an in the Taylor series for f(z) about z0 must be zero. Thus in neighborhood N0 since that Taylor series also represents f(z) in N0. 圖解 Theorem. If a function f is analytic throughout a domain D and f(z)=0 at each point z of a domain or arc contained in D, then in D. tch-prob

along real x-axis (an arc) Corollary: A function that is analytic in a domain D is uniquely determined over D by its values over a domain, or along an arc, contained in D. Example: along real x-axis (an arc) tch-prob

59. Behavior of f near Removable and Essential Singular Points Observation : A function f is always analytic and bounded in some deleted neighborhood of a removable singularity z0. tch-prob

Assume f is not analytic at z0. Thm 1: Suppose that a function f is analytic and bounded in some deleted neighborhood of a point z0. If f is not analytic at z0, then it has a removable singularity there. Pf: Assume f is not analytic at z0. The point z0 is an isolated singularity of f and f(z) is represented by a Laurent series If C denotes a positively oriented circle tch-prob

is satisfied at some point z in each deleted neighborhood Thm2. Suppose that z0 is an essential singularity of a function f, and let w0 be any complex number. Then, for any positive number , the inequality (a function assumes values arbitrarily close to any given number) (3) is satisfied at some point z in each deleted neighborhood tch-prob

Since z0 is an isolated singularity of f. There is a Pf: Since z0 is an isolated singularity of f. There is a throughout which f is analytic. Suppose (3) is not satisfied for any z there. Thus is bounded and analytic in According to Thm 1, z0 is a removable singularity of g. We let g be defined at z0 so that it is analytic there, becomes analytic at z0 if it is defined there as But this means that z0 is a removable singularity of f, not an essential one, and we have a contradiction. tch-prob

tch-prob