# Cauchy-Riemann Equations.

Recall the definition of the derivative of a function at a point :

Denote: , , and , where , , , , , , and are real.

Suppose that ; thus we have:

Suppose that ; thus we have:

Hence we have the so-called Cauchy-Riemann Equations:

which can be wriiten in the following form, with a notation frequently used in Calculus:

Theorem 3.3.1   If is derivable at , then and verify the Cauchy-Riemann Equations at .

Example 3.3.2   Let . As a polynomial function, is derivable over the whole of .

Let us check the Cauchy-Riemann equations. Denote . Then we have:

It follows that:

at every point in the plane, i.e. Cauchy-Riemann equations hold everywhere.

Example 3.3.3   Let . If , where are real numbers, then:

Let us check at which points the Cauchy-Riemann equations are verified. We have: , , and .Cauchy-Riemann equations are verified if, and only if, , i.e. . The only point where can be differentiable is the origin.

There is a kind of inverse theorem:

Theorem 3.3.4   If verifies the Cauchy-Riemann Formulas at and if the partial derivatives of and are continuous at , the is derivable at and .

Example 3.3.5   Let ; the function is derivable at any point and .

If , then . Then: , , and . These partial derivatives verify the C-R equations.

By that way, we have a new proof of the differentiability of at every point.

Example 3.3.6   Let , for any . We work as in the previous examples:

We compute the first partial derivatives:

We solve Cauchy-Riemann equations:

The subset of the plane where can be differentiable is the union of the two coordinate axes. As the first partial derivatives of and are continuous at every point in the plane, is differentiable at every point on one of the coordinate axes.

Cauchy-Riemann equations in polar form:

Instead of , write , where .

Noah Dana-Picard 2007-12-24