Page 93 - Calculus with Complex Numbers

3 D erivatives
If c = reio = .v + iy, then

X
log c = log r + io = 1 lo (.x2 + y2) + i tan- 1 -
z g .
A
ut 1 l (x2 + y2) gt x, y) = tan-l -
X
. x, y) - z og . , . .
A

ut.'r, y) = .r(.r2 + y2 - 2), tlt.x, y) = -y@2 + ,2 - 2).

é) u o o é) u é) ??
- = 3.:7* + yz' - 2, - = 2.xy, - = -2.xy,
é).x é) y é).x
é) u é) t?
- = - - for a1l .x y.
J )' J.Y '
é) u é) t?
- = - only when
9.:7 é) y
3.x2 + ,2 - 2 = - 2 - 3y2 + (2
.x
which simplilies to

2
.r .y y2 . j .

For Iz I= 1 we have

lf f (z) is real valued, then t? (x , y) = 0. Therefore,

é) u é) t? bu 9 t?
= = 0, = - = 0.
9.:7 é) y èy 9.1
Therefore u (x , y) = corlstant.
ez sin z = z + .:2 + c3/3 - c5/30 + . . . .