User:Domar1973/Calculus on complex coordinates
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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
[edit]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
[edit]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
[edit]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