1550251515-Classical_Complex_Analysis__Gonzalez_

564 Chapters

2. 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. Let lei I = lc2 I = · · · = !cm I = 1. Show that the series

00

L

-c 1 ( -n + .. ·+c -n) z n

n 1 m

n=l

converges everywhere on lzl = 1 except at the points c1, c2, ... , Cm.

4. (a) Use the expansion (8.3-11) and Abel's theorem to show that

ln 2 = 1 - 1/2 + 1/s -1/4 + ...

(b) Use the expansion Arctanz = z - z^3 /3 + z^5 /5 - .. · and Abel's

theorem to show that

~ = 1 -1;3 + 1;5 -1;7 + ....

4

5. Use Abel's formula of summation by parts to show that the series
   I:~ 1 akgk(z) converges uniformly on a set B if:
   (1) The partial sums Sn(z) = I:~=l gk(z) are bounded for all n and
   all z E B (i.e., uniformly bounded in B).
   (2) The real sequence {ak} approaches zero monotonically from some
   value of k on (Dirichlet's test). ·
   Apply this test to show that the series
   00 k 00 iklJ
   I:~ =l:y
   k=l k=l

converges uniformly on every interval 8:::; 0:::; 27r - 8, where 0 < 8 < ?r,

and so convergent for all 0 =fa 2n?r. From this result, the expansion for

Log(l - z) and Abel's limit theorem, deduce that

and

f: sin kO _ 7r - 0

k=l _k_ - -2-

L.J ~ coskkO __ - Log(2 sin %0)

k=l

for 0 < 0 < 27r.

6. In a similar fashion, show that

~ sin(2k + 1)0 = 1 /

~ 2k+ 1 4 71"

(0<0<1r)

and