1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 565 ~ cos(2k + 1)() = 1; 1 I 1; e1 L..J 2k 1 2 n cot 2 k=O + (o < 1e1 < 7r) *7. S ...

566 Chapter^8 Example f(z) = tanz is meromorphic in lzl < 2. Its poles in this region are at z = ±%7r (zeros of cotz = 1/tanz ...

Sequences, Series, and Special Functions 567 Proof L'et M be the set of such zeros. We have f(M) = {O} and M = f-^1 {O} n D. Sin ...

568 Chapters Hence lf(z) - f(zo)I:::; lz - zol Thus for any given f > 0 we have lf(z)-f(zo)I < f provided that lz-zol < ...

Sequences, Series, and Special Functions which in view of the hypothesis, reduces to where 1 f(z) = j<m~~zo) (z - zo)m + ~:+~ ...

570 Chapter^8 Theorem 8.24 (Identity Principle for Analytic Functions). If two func- tions f and g are analytic in a region R an ...

Sequences, Series, and Special Functions 571 this particular case we can be much more specific about the number of its zeros, it ...

572 Chapter^8 (See Fig. 8.5.) Since imaginary parts of reciprocal complex numbers are opposite in sign, we have b Im--<0 z - ...

Sequences, Series, and Special Functions 573 x Fig. 8.6 Theorem 8.27 (Jensen-Walsh). Every nonreal zero of the derivative of a p ...

574 Chapter^8 lies outside all Jensen circles, P'( ) n 1 Im P(z) =Im L --= -2yK Z k=l Z - CXk K being positive. Hence if z is no ...

Sequerices, Series, and Special Functions 575 (b) Show that the distance from the origin to the nearest zero off is not rlaol le ...

576 Chapter^8 Since f(z) is not constant in G there is a first coefficient after ao, say ak(k 2:: 1), which is not zero. Let a 0 ...

Sequences, Series, and Special Functions 577 Proof Similar to that ~f Theorem 6.29. Remarks I. Some authors call Theorem 8.29 th ...

578 Chapter^8 and it is clear that the maximum is attained at t = 0 and the minimum at t = Tr. Hence max lf(z)I = 3 lzl:9 and mi ...

Sequences, Series, and Special Functions 8.12 MAXIMUM AND MINIMUM PRINCIPLES FOR REAL HARMONIC FUNCTIONS 579 In Section 6.6 we h ...

580 Chapters the integrand on the right is less than or equal to M, in fact, strictly less than M on some arcs of C containing t ...

Sequences, Series, and Special Functions 581 min_[u(x,y)-U(x,y)] = min [u(x,y)-U(x,y)] = 0 (x,y)EG (x,y)E8G so that u(x,y) = U(x ...

582 Chapters 11. If f is a nonconstant entire function, show that there exists an arc ""(: z = z(t), 0:::; t < oo, along whic ...

Sequences, Series, and Special Functions 583 then g(z) is analytic in izl < 1, and g(z) = f(z) z for z '=f 0 and g(O) = f'(O) ...

584 Chapters Geometric interpretation. If we let w = f(z), we see from lwl :::; lzl :::; r < 1 that, under the assumptions of ...