Let

(1) 
where

(2) 
so

(3) 
The total derivative of with respect to may then be computed as follows.
so
and

(8) 
In terms of and , (8) becomes
Along the real, or xAxis,
, so

(10) 
Along the imaginary, or axis,
, so

(11) 
If is Complex Differentiable, then the value of the derivative must be the same for a given , regardless of its
orientation. Therefore, (10) must equal (11), which requires that

(12) 
and

(13) 
These are known as the CauchyRiemann equations. They lead to the condition

(14) 
The CauchyRiemann equations may be concisely written as
In Polar Coordinates,

(16) 
so the CauchyRiemann equations become
If and satisfy the CauchyRiemann equations, they also satisfy
Laplace's Equation in 2D, since

(19) 

(20) 
By picking an arbitrary , solutions can be found which automatically satisfy the CauchyRiemann equations and
Laplace's Equation. This fact is used to find socalled Conformal Solutions to
physical problems involving scalar potentials such as fluid flow and electrostatics.
See also Cauchy Integral Theorem, Conformal Solution, Monogenic Function,
Polygenic Function
References
Abramowitz, M. and Stegun, C. A. (Eds.).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 17, 1972.
Arfken, G. ``CauchyRiemann Conditions.'' §6.2 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 3560365, 1985.
© 19969 Eric W. Weisstein
19990526