Let
 | (1) |
where
 | (2) |
so
 | (3) |
The total derivative of 
with respect to 
is then
 |  |  | (4) |
 |  |  | (5) |
In terms of and 
, (5) becomes
 |  |  | (6) |
 |  |  | (7) |
Along the real, or x-axis, 
, so
 | (8) |
Along the imaginary, or y-axis, 
, so
 | (9) |
If 
is complex differentiable, then the value of the derivative must be the same for a given 
, regardless of its orientation. Therefore, (8) must equal (9), which requires that
 | (10) |
and
 | (11) |
These are known as the Cauchy-Riemann equations.
They lead to the conditions
 |  |  | (12) |
 |  |  | (13) |
The Cauchy-Riemann equations may be concisely written as
 |  |  | (14) |
 |  |  | (15) |
 |  |  | (16) |
 |  |  | (17) |
where is the complex conjugate.
If 
, then the Cauchy-Riemann equations become
 |  |  | (18) |
 |  |  | (19) |
(Abramowitz and Stegun 1972, p. 17).
If and 
satisfy the Cauchy-Riemann equations, they also satisfy Laplace’s equation in two dimensions, since
 | (20) |
 | (21) |
By picking an arbitrary 
, solutions can be found which automatically satisfy the Cauchy-Riemann equations and Laplace’s equation. This fact is used to use conformal mappings to find solutions to physical problems involving scalar potentials such as fluid flow and electrostatics.
