1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 525 Proof Letting Fn(z) = f1(z)+· · + fn(z), the functions Fn(z) (n = 1, 2, ... ) satis ...

526 Chapter 8 Proof Let B be any compact subset of A. As in the proof of Theorem 8.2, let p__ = d(B,BA), and for any z E B consi ...

Sequences, Series, and Special Functions 527 f(z, t) is a function of the complex variable z on the open set A, and of the real ...

528 Chapter 8 so that under such conditions the reversal of the order of integrations is permissible. In fact, if z 0 is a fixed ...

Sequences, Series, and Special Functions 529 = J [l 00 f((, t) dt] d( :S €L(C) c Hence j [fo 00 t((,t)dt] d( = J~foT [iat((,t)d( ...

530 Chapter^8 that lz --al = p < R. For the case R = oo, let z be an arbitrary point of the complex plane, i.e., such that lz ...

Sequences, Series, and Special Functions 531 IC -zl = l(C -a) -(z - a)I 2:: IC -al -lz -al = 'T' - p. Hence, applying Darboux's ...

532 Chapters The uniqueness of the expansion (8.3-5) follows from the identity prin- ciple for power series (Theorem 4.23). This ...

Sequences, Series, and Special Functions 533 If n = 2m the coefficients vanish, and if n = 2m + 1, the coefficients reduce to 1 ...

534 Chapters by making use of Corollary 8.1. To expand f(z) = (1 + z)P in powers of z, where pis a given complex number (in par ...

Sequences, Series, and Special Functions 535 7. To expand f(z) =Log z in powers of z - (-2 + i). This function is analytic in D ...

536 series expansion, namely, 1 = 1-x2 + x4 - ... + (-ltx2n + ... 1 +x^2 Chapter 8 converges only for lxl < 1, i.e., for x in ...

Sequences, Series, and Special Functions 537 8.4 Operations with Power Series In the preceding section we have seen that there a ...

538 Chapter 8 Note 'rhe radius of convergence of 2:::"= 0 (aan + f3bn)zn can be.greater than R = min(Ri, R 2 ). For instance, we ...

Sequences, Series, and Special Functions 539 zeros of g (if I< is not empty), and p = oo if g has no zeros in C. Then · f(z) ...

540 Chapter^8 from which the unknown coefficients can be obtained successively. We find that ao Co=-, bo _ ]__ Ibo ao I C1 - b2 ...

Sequences, Series, and Special Functions 541 Then the coefficients in every column above form a convergent series, and if we let ...

542 Chapter^8 and since L: rm converges for 0 :::; r < 1, the uniform convergence of (8.4-7) on lzl :::; r follows from the W ...

Sequences, Series, and Special Functions 543 Fn(z) = An[f(z) - ao]n are analytic in the same disk lzl < r, it follows from Th ...

544 Chapter^8 Proof Since a 1 = f'(O) f:. 0, the existence and analyticity of a single-valued inverse in a neighborhood of the o ...