1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 2. lf(z)I ~ 1 for lzl < 1 3. J(c) = 0, where 0 < lei < 1 Then IJ(z)I ~I; ~c: I ...

586 Chapter^8 lf'(e)I:::; 1 -lle12 Equality occurs iff F(z') = az', or f(z) = a[(z -e)/(1-cz)] with !al= 1. The following theore ...

Sequences, Series, and Special Functions 587 Assuming that f is not a constant function, first we consider the case f(O) = 0. Si ...

588 Chapter^8 Exercises 8.5 1. If f is analytic in lzl < 1 and such that IJ(z)I < 1, show that (G. Pick [28]): (a) I f(z)- ...

Sequences, Series, and Special Functions 589 *6. 00 (a) Let f(z) = L CnZn for lzl < R, and let A(r) = maxRef(z), where n=O Jz ...

590 Chapter^8 and r2 log(Ma/M1) r2 logM 2 :=:; -alog - +logM1 = l ( / ) log - +logM1 ri og ra ri ri ra r2 ra ( log - log M 2 :=: ...

Sequences, Series, and Special Functions 591 (a) Prove that a convex function e over [xi, X2] is nece~sarily continuous in (x1, ...

592 Chapter 8 1. If r == 0, the series (8.15-2) converges absolutely for every finite value of(. Hence the series (8.15-1) conve ...

Sequences, Series, and Special Functions 593 A Laurent series is often written in the form +oo '""" ~ A m ( z - a )m = · .. + A_ ...

594 Chapter^8 Fig. 8.8 where r < R, then the function f(z) = f1(z) + fz(z) is said to be the function defined by the Laurent ...

Sequences, Series, and Special Functions 595 Fig. 8.9 For t E Ci we have 1 1 =------ t - z. (t - a) - (z - a) 1 1 ~ (z - a)n (8. ...

596 Chapter^8 where An= _1 J f(t)dt 27fi (t -a)n+l (8.18-7) C1 Similarly, for t E C 2 we have 1 1 t-z (t-a)-(z-a) 1 1 00 (t-ar-^ ...

Sequences, Series, and Special Functions 597 Substitution of (8.18-6) and (8.18-10) into (8.18-3) gives 00 00 f(z) = L An(z-at+ ...

598 Chapter^8 with r' ~ lz -al ::::; R'. For any given f > 0, there exists a positive integer N such that for n ~ N and all z ...

Sequences, Series, and Special Functions 599 Proof Suppose that we have +oo f(z) = I: Am(z - ar (8.18-14) m=-oo and also +oo f(z ...

600 Now 1 z-2 1 1 oo 2n-l -; 1 - 2; z = 'E ---;-;;- n=l valid for 12/zl < ll, or lzl > 2, and 1 1 1 00 Zn 4 - z = 4 1 - z/ ...

Sequences, Series, and Special Functions 601 = 4~i [/ (i _ :;tm+i +I (4-:;tm+i] where r is any circle with center at the origin ...

602 Chapter^8 Fig. 8.10 (Fig. 8.10). Since z z^2 zn ez = 1 + - + - + · · · + - + · · · 1! 2! n! which converges for any finite v ...

Sequences, Series, and Special Functions 603 ~ ~ 2 ~ j cos(nB-tsinB)dB- 2 ~ 1·sin(nB-tsinB)dB -~ -~ The integrand of the first i ...

604 Chapter^8 Find the Laurent expansion of 1 f(z) = (z - l)(z - 2)(z - 3) in each of the following regions: (a) 1 < lzl &l ...