Complex Differentiability

Complex Differentiability

-Differentiability

Suppose is open, a function is -differentiable at if exists, in which case we call the limit . Equivalently, is -differentiable if there exists a complex number such that

e.g. is -differentiable but not -differentiable.

The Cauchy-Riemann Equations

Cauchy-Riemann Equations

Let be open, let , and let . Write . Then the Cauchy-Riemann equations are

Dolbeaut Operators

The Dolbeaut operators for some complex functions are defined as:

e.g. This agrees with how we think differentiating with respect to should work, and indeed the same goes for applying and to polynomials in and . We will see for one thing that is -differentiable only at by the following theorem:

Theorem

Let be open, let , and let . Write . Then the following are equivalent:

• is -differentiable at and .
• is -differentiable at and the Jacobian .
• is -differentiable at , and we have the Cauchy-Riemann equations hold.
• is -differentiable at , and . In this case .

Proof The last three arguments are equivalent simply by definition of Jacobian. To see the first two are equivalent, suppose , we just check this is equivalent to the Jacobian in terms of multiplication:and accordingly in , we have $$\begin{pmatrix}a\Delta x-b\Delta y\b\Delta x+a\Delta y\end{pmatrix}=\begin{pmatrix}a&-b\b&a\end{pmatrix}\begin{pmatrix}\Delta x\\Delta y\end{pmatrix}$$$\square$

e.g. Consider . Cleary is -differentiable. Now check that the Cauchy-Riemann equation holds: So is -differentiable everywhere.

Corollary

-differentiable functions are continuous.

Proof -differentiability implies -differentiability, and -differentiability implies continuity.