Cauchy-Riemann Equations.

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

Suppose that ; thus we have:

Suppose that ; thus we have:

Hence we have the so-called

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.

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:

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.

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