Complex Variables and Applications by James Brown & Ruel Churchill

Author:James Brown & Ruel Churchill
Language: eng
Format: epub
Publisher: UNKNOWN
Published: 2018-04-25T00:00:00+00:00

∗For a proof of Picard’s theorem, see Sec. 51 in Vol. III of the book by Markushevich, cited in sec. 72 Exercises 243 In the remaining sections of this chapter, we shall develop in greater depth the theory of the three types of isolated singular points just described. The emphasis will be on useful and efficient methods for identifying poles and finding the corresponding residues.

EXERCISES 1. In each case, write the principal part of the function at its isolated singular point and determine whether that point is a pole, a removable singular point, or an essential singular point:

(a) zexpz2 11 ; (b)1+z; (c) sinz; (d) cosz; (e)(2−z)3.z z z

Show that the singular point of each of the following functions is a pole. Determine2.

the order m of that pole and the corresponding residue B. (a) 1− coshz; (b) 1− exp(2z); (c) exp(2z).z3 z4 (z− 1)2 Ans . (a) m = 1,B=−1/2; (b) m = 3,B=−4/3; (c) m = 2,B = 2e2. 3. Suppose that a function f is analytic at z0, and writeg(z) =f(z)/(z−z0). Show that (a) iff(z0) = 0, then z0 is a simple pole of g, with residuef(z0); (b) iff(z0) = 0, then z0 is a removable singular point of g.

Suggestion: As pointed out in Sec. 57, there is a Taylor series forf(z) about z0 since f is analytic there. Start each part of this exercise by writing out a few terms of that series.

4. Use the fact (see Sec. 29) that ez=−1 when

z = (2n+ 1)π i (n = 0,±1,±2,...)

to show that e1/z assumes the value −1aninfinite number of times in each neighborhood of the origin. More precisely, show that e1/z=−1 when

i z=−(2n+ 1)π (n = 0,±1,±2,...); then note that if n is large enough, such points lie in any given ε neighborhood of the origin. Zero is evidently the exceptional value in Picard’s theorem, stated in Example 5, Sec. 72.

5. Write the function8a3z2

f(z) =(z2 +a2)3 (a > 0)

as3z2 f(z)

=

φ(z)

(z−ai)3 where φ(z) =8a .(z+ai)3 Point out whyφ(z) has a Taylor series representation about z =ai, and then use it to show that the principal part of f at that point is

φ (ai)/2+ φ (ai) + φ(ai) =− i/2 − a/2 a2i (z−ai)2 − (z−ai)3.z−ai (z−ai)2 (z−ai)3 z−ai

73. RESIDUES AT POLES When a function f has an isolated singularity at a point z0 , the basic method for identifying z0 as a pole and finding the residue there is to write the appropriate Laurent series and to note the coefficient of 1/(z−z0). The following theorem provides an alternative characterization of poles and a way of finding residues at poles that is often more convenient.

Theorem. An isolated singular point z0 of a function f is a pole of order m if and only iff(z) can be written in the form

(1) f(z)φ(z) , =(z−z0)m

whereφ(z) is analytic and nonzero at z0 . Moreover, (2) Res f(z) =φ(z0) if m = 1

z=z0

and

(3) Resf(z) = φ(m−1)(z0) if m ≥ 2. z=z0 (m− 1)! Observe that expression (2) need not have been written separately since, with the convention that φ(0)(z0) =φ(z0) and 0! = 1, expression (3) reduces to it when m = 1.

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