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Cauchy-Riemann Conditions

A function of a complex variable ,  defined and finite in a neghourhood of a point , is said to be differentiable at this point, if the limit

exists, is finite and the same no matter how approaches zero (or approaches ) in the complex plane.

Though the definition is formally identical with that used in real analysis, the condition of differentiability in the complex plane is much stronger than in the real domain analysis. For instance, the function as a function of two real variables is differentiable at if both functions and are differentiable at this point.  

Example: The function is differentiable as a function of two real variables in the whole plane .  As a function of one  complex variable it is differentiable only at (see below).

The following result reduces differentiation in the complex domain to the differentiation of real functions. The conditions can already be found in the work of d’Alembert and Euler:

Cauchy-Riemann conditions: In order that the function

is differentiable at the point , it is necessary and sufficient that both its real an imaginary parts, and , have a total differential at and satisfy equations

Equivalently, that the function possesses a total differential and satisfies the equation

If this condition is satisfies, then

Example: The function   satisfies Cauchy-Riemann conditions at no point.

Example: The function satisfies Cauchy-Riemann conditions at point only.

Example: Let

and . This is the graph of this function.

The Cauchy-Riemann conditions are satisfied at . Moreover exists when tends to along an arbitrary straight line passing through the point .  

Nevertheless, the function is not differentiable at .

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