1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions Hence, z = Log () 1 + w = w - l2 -w + -l3 w - -^14 w + · · · 2 3 4 which converges for ...

546 Chapter^8 which is precisely the product of the two series 00 1 +La~ Zn n=l n. and Hence the use of the symbolic method is j ...

Sequences, Series, and Special Functions 547 B2 2 Ba 3 =1+ -z + -z + ... 2! 3! (8.5-4) But g(z) = (z/2)cothz/2 is an even functi ...

548 Chapter^8 and letting z 0 ---+ 0 we have . = 22nB · smz "C )n 2n 2 n Log -z-= ~ -l (2n)(2n)! z ' lzl < 7f (8.5-11) Note W ...

Sequences, Series, and Special Functions 549 The numbers E 2 n [or the ( -1 r E 2 n] are called the Euler numbers. They were int ...

550 By using multiplication of series, prove the following. (a) (1 + z + z^2 + · · ·)^2 = 1+2z + 3z^2 + · · ·, lzl < 1 (b) 2 ...

Sequences, Series, and Special Functions 551 show that the Bernoulli numbers also satisfy the following symbolic equations. (a) ...

552 Suppose that f(z) = L:::o anzn, JzJ < 1, and that Jf(z)J:::; 1-1JzJ2 Prove that Jani <^1 / 2 (n + 2)e Chapter 8 Let ...

Sequences, Series, and Special Functions 553 Theorem 8.12 Let f(z) = u(x, y) + iv(x, y) and suppose that the func- tions u and v ...

554 Hence (8.6-4) can be written in the form n 1 f(z + D.z) = f(z) + L kl (fzD.z + f-zD."i)(k) +Rn k=l Chapter 8 (8.6-5) This is ...

Sequences, Series, and Special Functions Hence if M = max( Mi, M 2 ), we obtain 2n+2M IRnl < (n + l)! rn+l 555 (8.6-7) since ...

556 Chapter^8 coefficients an. In fact, either case may occur, as the following examples show. Examples 1. If .f(z)=- tc^1 Log(l ...

Sequences, Series, and Special Functions 557 so that n+p-1 + L (ak - ak+i)lzlk+l + an+plzjn+p+l k=n+l For jzj = 1 but z f= 1, we ...

558 Chapters Proof As a preliminary step we shall prove the so-called Abel's formula of summation by parts; namely, if {bn} and ...

Sequences, Series, and Special Functions 559 The right-hand side of (8.7-4) can be made as small as we please by taking n suffic ...

/ 560 Hence 11-zl < k 1-lzl - Chapters for any z in (}' except the vertex A. Consequently, if we let z ~ 1 along any path, ul ...

Sequences, Series, and Special Functions 561 All three series converge for z = 1, so they converge absolutely for JzJ < 1. Si ...

562 Chapter^8 and write 00 Note Any theorem which states that if a sequence converges to a value, then some average of the seque ...

Sequences, Series, and Special Functions 563 as N ---+ oo. Now, given e > 0, choose N (i.e., lzl sufficiently close to 1) suc ...

564 Chapters Show that the series f(z) = I:~=l nzn-l diverges at all points of lzl = 1. What kind of point is z = 1 for f? 3. ...