Page 410 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 401

d z þ z z z x þ iy þ x þ i y x þ iy
z z ¼ lim ¼ lim
dz z!0 z x!0 x þ i y
y!0
x i y
¼ lim x iy þ x þ i y ðx iyÞ ¼ lim
x!0 x þ i y x!0 x þ i y
y!0 y!0
x
If y ¼ 0, the required limit is lim ¼ 1:
x!0 x
i y
If x ¼ 0, the required limit is lim ¼ 1:
y!0 i y
These two possible approaches show that the limit depends on the manner in which z ! 0, so that the
derivative does not exist; i.e., z is nonanalytic anywhere.
z

1 þ z dw
, find .(b) Determine where w is nonanalytic.
1 z dz
16.6. (a)If w ¼ f ðzÞ¼
1 þ z
1 þðz þ zÞ
dw 1 z 2
Method 1: ¼ lim 1 ðz þ zÞ ¼ lim
dz z!0 z z!0 ð1 z zÞð1 zÞ
ðaÞ
2
¼ 2 provided z 6¼ 1, independent of the manner in which z ! 0:
ð1 zÞ
Method 2. The usual rules of differentiation apply provided z 6¼ 1. Thus, by the quotient rule for
differentiation,
d d

d 1 þ z ð1 zÞ dz ð1 þ zÞ ð1 þ zÞ dz ð1 zÞ ð1 zÞð1Þ ð1 þ zÞð 1Þ 2
dz 1 z ¼ 2 ¼ 2 ¼ 2
ð1 zÞ ð1 zÞ ð1 zÞ
(b) The function is analytic everywhere except at z ¼ 1, where the derivative does not exist; i.e., the function
is nonanalytic at z ¼ 1.

16.7. Prove that a necessary condition for w ¼ f ðzÞ¼ uðx; yÞþ ivðx; yÞ to be analytic in a region is that
@u @v @u @v
the Cauchy-Riemann equations ¼ , ¼ be satisfied in the region.
@x @y @y @x
Since f ðzÞ¼ f ðx þ iyÞ¼ uðx; yÞþ ivðx; yÞ,we have

f ðz þ zÞ¼ f ½x þ x þ ið y þ yފ ¼ uðx þ x; y þ yÞþ ivðx þ x; y þ yÞ
Then
lim f ðz þ zÞ f ðzÞ ¼ lim uðx þ x; y þ yÞ uðx; yÞþ ifvðx þ x; y þ yÞ vðx; yÞg
z!0 z x!0 x þ i y
y!0
If y ¼ 0, the required limit is

@u @v
lim uðx þ x; yÞ uðx; yÞ þ i vðx þ x; yÞ vðx; yÞ þ i
x!0 x x ¼ @x @x
If x ¼ 0, the required limit is

1 @u @v
lim uðx; y þ yÞ uðx; yÞ þ vðx; y þ yÞ vðx; yÞ ¼ þ
y!0 i y y i @y @y
If the derivative is to exist, these two special limits must be equal, i.e.,
@u @v 1 @u @v @u @v
þ i ¼ i
@x @x ¼ i @y þ @y @y þ @y