The Real And Complex Number Systems

ux, y
(1) arctany/x, ifx 0, y R
(2) /2, if x 0, y 0
(3) arctany/x , ifx 0, y 0
(4) arctany/x , ifx 0, y 0
(5) /2, if x 0, y 0.
and
v x v y 0.
So, we know that by (1)-(5), for x, y 0, 0
y
u x
x 2 y 2
and for x, y x, y : x 0, y 0, wehave
u y x
x 2 y . 2
Hence, we know that no z makes Cauchy-Riemann equations hold.
Remark: We can give the conclusion as follows:
y
arg z x
for x, y 0, 0
x 2 y 2
and
arg z y
x for x, y x, y : x 0, y 0.
x 2 y2 (f) Log z u iv, where
ux, y logx 2 y 2 1/2 and vx, y argx 2 y 2 1/2 .
Since
u x x
x 2 y and u y
2 y
x 2 y 2
and
y
v x
x 2 y for x, y 0, 0 and v 2 y x for x, y x, y : x 0, y 0,
x 2 y2 we know that all z except real z 0 make Cauchy-Riemann equations hold.
Remark: Log z is differentiable on C x, y : x 0, y 0 since Cauchy-Riemann
equations along with continuity of u x iv x ,andu y iv y .
(g) e z2 u iv, where
ux, y e x2 y 2 cos2xy and vx, y e x2 y 2 sin2xy.
So,
u x v y 2e x2 y 2 xcos2xy ysin 2xy for all z x iy.
and
u y v x 2e x2 y 2 ycos2xy xsin 2xy for all z x iy.
Hence, we know that all z make Cauchy-Riemann equations hold.
(h) Since z e Log z ,ande z is differentiable on C, we know that, by the remark of (f),
we know that z is differentiable for all z except real z 0. So, we know that all z except
real z 0 make Cauchy-Riemann equations hold.
( In part (h), the Cauchy-Riemann equations hold for all z if is a nonnegative integer,