Derivative of a Complex Function
The derivative of a complex function ω=f(z) = u(x, y ) + i v(x, y ), where z=x+i y ,
is defined in the exact same way as that of a real function y=f(x). That is,
dω
dz=f′(z) = lim∆z→ 0 f(z+ ∆ z)−f(z)
∆z
dω
dz=f′(z) = lim∆z→ 0 u(x+ ∆ z, y + ∆ y) + i v(x+ ∆ x, y + ∆ y)−[u(x, y ) + i v (x, y )]
∆x+i∆y(12.178)if this limit exists. In the definition of a derivative, equation (12.178), the limit must
be independent of the path taken as ∆ztends toward zero.
Consider the points z=x+i y and z+ ∆ z= (x+ ∆ x) + i(y+ ∆ y)in the z−plane as
illustrated in the figure 12-11.
Figure 12-12. Delta z approaching zero along different paths.
If the limit in equation (12.178) approaches zero along the path 1 of figure 12-12,
then the definition given by equation (12.178) becomes, after first setting ∆x= 0
dω
dz=f′(z) = lim
∆y→ 0u(x, y + ∆ y) + i v (x, y + ∆ y)−[u(x, y ) + i v(x, y )]
i∆y
dω
dz=f′(z) = lim∆y→ 0 u(x, y + ∆ y)−u(x, y )
i∆y+ lim∆y→ 0 v(x, y + ∆ y)−v(x, y )
∆y
dω
dz=f′(z) = −i∂u
∂y+∂v
∂ySee page 158, Volume I
(12.179)