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

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Choosing orthogonal coordinates for the euclidean plane   and   (which can be also thought of as real and imaginary axis of the complex plane  ), we introduce complex coordinates   and   through

 

By doing so, one is regarding   and   as independent variables, and any function on the plane becomes a function on  , with the understanding that the hyperplane defined by   is the "actual" complex plane.

We can define then the 1-forms

 

and the "partial derivatives" (tangent vectors)

 

with the properties

 

and

 

A function   in the complex plane can be regarded as a function in  :

 

by replacing   in the arguments for  .

Observe that   and   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

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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  . Using the above definitions a little manipulation leads to write the Cauchy Riemann equations in the form

 

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

 

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

Observe that, if the function   is holomorphic, then

 

Real two dimmensional calculus in complex coordinates

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Clasical theorems of calculus on   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

Higher dimmensions

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References

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Further reading

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