User:Domar1973/Calculus on complex coordinates

In physics, in particular in Conformal field theory and String Theory, it is widely used a "complexification" of coordinates in terms of which both real and complex analysis is cast. This permits to employ the power of complex analysis to system formulated on euclidean space but possessing strong symmetries like conformal symmetry.

Complexification of the euclidean plane
Choosing orthogonal coordinates for the euclidean plane $$x$$ and $$y$$ (which can be also thought of as real and imaginary axis of the complex plane $$C$$), we introduce complex coordinates $$z$$ and $$\bar z$$ through

$$ z = x + i y \qquad \bar z = x - i y $$

By doing so, one is regarding $$z$$ and $$\bar z$$ as independent variables, and any function on the plane becomes a function on $$C^2$$, with the understanding that the hyperplane defined by $$z^* = \bar z$$ is the "actual" complex plane.

We can define then the 1-forms

$$dz = dx + i dy \qquad d \bar z = dx - i dy$$

and the "partial derivatives" (tangent vectors)

$$\partial = \frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \qquad \bar{\partial} = \frac{\partial}{\partial \bar{z}} = \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) $$

with the properties

$$\partial z = 1 \qquad \partial \bar z = 0$$

and

$$\bar{\partial} z = 0 \qquad \bar{\partial} \bar z = 1$$

A function $$f(z): C \rightarrow C$$ in the complex plane can be regarded as a function in $$C^2$$:

$$f(z,\bar z): C^2 \to C$$

by replacing $$\mbox{Re}z=z+\bar z \qquad \mbox{Im}z=-i(z-\bar z)$$ in the arguments for $$f$$.

Observe that $$\partial$$ and $$\bar{\partial}$$ exist whenever the euclidean partial derivatives does, so up to this point concepts of complex analysis have not appeared yet. This can be regarded as an alternative description of the plane and mere notation.

Holomorphic and antiholomorphic functions
Since the Cauchy-Riemann equations are relations between real partial derivatives, they can be cast in this formalism. It is then when its strength shows.

Let $$f(z):C \rightarrow C$$. Using the above definitions a little manipulation leads to write the Cauchy Riemann equations in the form

$$\bar{\partial} f(z,\bar z) = 0 $$

In the same sense, the so called antiholomorphic functions (that is, functions holomorphic in terms of $$z^*$$) are those functions fullfilling the condition

$$\partial f(z,\bar z) = 0 $$

This is the precise meaning of the phrase often found in physics literature "Holomorphic functions are functions that do not depend on $$\bar z$$".

Observe that, if the function $$f$$ is holomorphic, then

$$\frac{d}{dz} f(z) = \partial f(z,\bar z) |_{\bar z = z^*} $$

Real two dimmensional calculus in complex coordinates
Clasical theorems of calculus on $$R^2$$ are usually recast in complex coordinates. It should be noticed that many expressions that look like complex analysis formulae are actually not, but simply a notation for real calculus. For instance

integral doble

delta de Dirac

teorema de la divergencia / Green