The Real And Complex Number Systems

ux, y ey e y cosx
2
and
vx, y ey e y sin x
.
2
(c) fz |z|,
Solution: Since|z| x 2 y 2 1/2 , we know that
ux, y x 2 y 2 1/2
and
vx, y 0.
(d) fz z,
Solution: Since z x iy, we know that
ux, y x
and
vx, y y.
(e) fz arg z, (z 0),
Solution: Since argz R, we know that
ux, y argx iy
and
vx, y 0.
(f) fz Log z, (z 0),
Solution: Since Log z log|z| i argz, we know that
ux, y logx 2 y 2 1/2
and
vx, y argx iy.
and
(g) fz e z2 ,
Solution: Since e z2
e x2 y 2 i2xy
, we know that
ux, y e x2 y 2 cos2xy
vx, y e x2 y 2 sin2xy.
(h) fz z ,( complex, z 0).
Solution: Since z e Log z , thenwehave(let 1 i 2 )
z e 1i 2 log|z|i arg z
e 1 log|z| 2 arg zi 2 log|z| 1 arg z
.
So, we know that
ux, y e 1 log|z| 2 arg z
cos 2 log|z| 1 arg z
e 1 logx 2 y 2 1/2 2 argxiy
cos 2 logx 2 y 2 1/2 1 argx iy