318 Chapter^6
Hence, the partial derivatives u.,, uy, v.,, Vy exist at (x,y), and they satisfy
the equations
Ux =Vy, Uy= -v.,.called the Cauchy-Riemann partial differential equations. t
As a by-product we have
J'(z) =A+ iB = u., +iv., =Vy - iuy
lf'(z)l2 = u; + v; = v~ + u~ = u.,vy - UyVx
=J(~) x,y
and if f'(z) -=/:-0, 'lj; = argf'(z), then
Ux Vy
cos'lj;= .IT= yJ .yJ .IT', .!, Vx -Uy
Sln'f'= - = --VJ VJ
(6.5-3)(6.5-4)
( 6.5-5)(6.5-6)Formula (6.5-4) gives two alternative ways to compute f'(z) in terms
of the partial derivatives of u and v, while (6.5-5) shows that the square
of the modulus of f'(z) equals the Jacobian of u and v with respect to x
and y at the point (x, y). Clearly, J = 0 iff f'(z) = 0. The Jacobian is
also denoted by &(u,v)/&(x,y) and by J1(z) whenever f = u +iv. The
equations in (6.5-3) may be obtained directly by assuming the existence
of f'(z) and computing limfa_. 0 (6w/6z) first by taking f1z = 6x, then
by taking 6z = i 6y.
- Now suppose that u and v are differentiable at ( x, y), and that the
equations (6.5-3) hold. Then we have
6u = u., 6x + Uy 6y + c
6v = v., 6x + Vy 6y + c^1
where c / 6z -+ 0 and c^1 / 6z -+ 0 as 6z -+ 0, and it follows that
D..w = 6u + i6v = (u., + iv.,)6x + (uy + ivy)6y + c + ic^1
By using (6.5-3) we obtain
and hence
D..w = ( u., + iv.,) 6x + ( -v., + iu.,) 6y + c + ic^1
= (u., +iv.,) 6x + (u., + iv.,)(i 6y) + c + ic'
= ( u., + iv.,) 6z + c + ic'6w. c. c^1
6z = u., + iv., + 6z + i 6z
tniscovered by D'Alembert in 1752.
(6.5-7)