If the limit in equation (12.178) approaches zero along the path 2 of figure 12-12,
then the definition given by equation (12.178) becomes, after first setting ∆y= 0
dω
dz =f′(z) = lim
∆x→ 0u(x+ ∆ x, y ) + i v (x+ ∆ x, y )−[u(x, y ) + i v (x, y )]
∆x
dω
dz =f′(z) = lim
∆x→ 0u(x+ ∆ x, y )−u(x, y )
∆x +i∆limx→ 0v(x+ ∆ x, y )−v(x, y )
∆x
dω
dz =f′(z) = ∂u
∂x +i∂v∂x See page 158, Volume I
(12.180)If the derivative of the complex function is to exist with the limit of equation
(12.178) being independent of how ∆zapproaches zero, then it is necessary that the
derivative results from the equations (12.179) and (12.180) must equal one another
or
dω
dz =f′(z) = ∂u
∂z +i∂v
∂x = −i∂u
∂y +∂v
∂y (12.181)Equating the real and imaginary parts in equation (12.181) one finds that a necessary
condition for the existence of a complex derivative associated with the complex
function given by ω=f(z) = u(x, y ) + i v(x, y )is that the following equations must be
satisfied simultaneously
∂u
∂x =∂v∂y and
∂v
∂x = −∂u
∂y (12.182)These simultaneous conditions are known as the Cauchy-Riemann equations for the
existence of a derivative associated with a function of a complex variable.
A function ω=f(z) = u(x, y ) + i v(x, y )which is both single-valued and differential
at a point z 0 and all points in some small region around the point z 0 , is said to be
analytic or regular at the point z 0.