Use of Laplace's equation in analytic function theory

[spaghetti3451]

I was wondering what analytic function theory means and how Laplace's equation comes in wide use in analytic function theory.

 

[Ssnow]

An analytic function $f$ (real or complex) is a function that locally is given as an convergent series

$$f(x)=\sum_{i=0}^{\infty}a_{i}(x-x_{0})^{i}.$$
It is in fact an infinitely differentiable function with a Taylor expansion centerd in some point $x_{0}$. Usually ther is an important relationship between $a_{i}$ and $f^{(i)}$ the $i$-derivative, expressed by $a_{i}=\frac{f^{(i)}(x_{0})}{i!}$. Examples of analytic functions are: polynomials (real or not), the exponential function, trigonometric functions ... an important fact in complex analysis is that analytic function are equivalent to holomorphic functions. A complex function is holomorphic if
$$\frac{\partial}{\partial \overline{z}}f=0$$

that is the same to say that the Cauchy Riemann Equations are satisfied. Now the complex Laplacian (in $\mathbb{C}$, for simplicity we consider only $1$ variables but it is possible to generalize ...) is

$$\Delta f= 4\frac{\partial^{2}}{\partial z \partial \overline{z}}f$$

that is $0$ if $f$ is holomorphic and then analytic... we call functions that $\Delta f=0$ harmonic functions and all harmonic functions are analytic ... (we can say that ''harmonic functions are real analogues to holomorphic functions'')

(For more details I suggest you some the beautiful book

''Teoria elementare delle funzioni analitiche di più variabili complesse''

of Salvatore Coen or

Henry Cartan, Elementary Theory of Analytic Functions of one or Several Complex Variables )

Hi,
Ssnow