The Real And Complex Number Systems

and
vx, y e 1 log|z| 2 arg z
sin 2 log|z| 1 arg z
e 1 logx 2 y 2 1/2 2 argxiy
sin 2 logx 2 y 2 1/2 1 argx iy .
(ii) Show that u and v satisfy the Cauchy -Riemanns equation for the following values
of z :Allz in (a), (b), (g); no z in (c), (d), (e); all z except real z 0in(f),(h).
So,
and
Proof: (a)sinz u iv, where
ux, y ey e y sin x
2
u x v y ey e y cosx
2
and vx, y ey e y cosx
2
for all z x iy
u y v x ey e y sin x
for all z x iy.
2
(b) cos z u iv, where
ux, y ey e y cosx
and vx, y ey e y sin x
2
2
So,
u x v y ey e y sin x
for all z x iy.
2
and
u y v x ey e y cosx
for all z x iy.
2
(c) |z| u iv, where
ux, y x 2 y 2 1/2 and vx, y 0.
So,
u x xx 2 y 2 1/2 v y 0ifx 0, y 0.
and
u y yx 2 y 2 1/2 v x 0ifx 0, y 0.
So, we know that no z makes Cauchy-Riemann equations hold.
(d) z u iv, where
ux, y x and vx, y y.
So,
u x 1 1 v y .
So, we know that no z makes Cauchy-Riemann equations hold.
(e) arg z u iv, where
ux, y argx 2 y 2 1/2 and vx, y 0.
Note that
.
.