Analytical function

A analytical function is a function which can be expressed locally like a whole Série convergent. In Analyze complexes the important result is that the holomorphic functions are analytical.

Definition

That is to say $f: U \ to \ mathbb {C} \,$ a function with complex variable, where $U \,$ is an open of $\ mathbb {C} \,$ . One says that the function $f \,$ est analytical on $U \,$ so for all $a \ in U \,$ , there exists a continuation $(a_ {N}) \,$ de complex numbers and there exists a reality $r>0 \,$ such as for all $z \ in D (has, R) \,$ , i.e. for all $z \,$ in the disc (open) of center $a \,$ and of ray $r \,$ , presumedly included in $U \,$ , one a:

$f (Z) = \ sum_ {n=0} ^ {+ \ infty} a_ {N} \, (z-a) ^ {N}$

In other words, a function is analytical if it is developable in whole series in the vicinity of each point of sound together open of definition.

Properties of the analytical functions

an analytical function is holomorphic. Besides there exists reciprocal with this proposal, namely that all holomorphic Fonction on open is analytical on this one.

Moreover, one analytical function is infinitely derivable (with the complex direction, to see holomorphic Fonction) and the derivative N - ième in a point $a \ in U \,$ is $f^ {(N)}(a)=n \! \, a_ {N} \,$ with the notations given in the definition. This proves that the development of $f$ in whole series in the vicinity of each point $a$ of $U$ is single; it is still called development in Taylor series .

the whole of the analytical functions on open is a algebra: the sum and the product of analytical functions are analytical.

When it is defined, the made up one of analytical functions is analytical.

Any series whole of ray of convergence not no one defines on its disc of convergence an analytical function. It is not commonplace, because a whole series is a priori a development in the vicinity of only one point.

All polynomial function is analytical on $\ mathbb {C}$ : it is said that it is whole . Being given a polynomial function, the terms of its development in whole series in the vicinity of an unspecified point of $\ mathbb {C}$ are all null starting from a certain row.

Analytical functions: examples and counterexamples

the Exponential function given by $\ exp (Z) = \ sum_ {n=0} ^ {+ \ infty} \ frac {z^ {N}} {N!}$ is analytical on $\ mathbb {C}$ : it is a whole function.

the function $f: \ mathbb {C} ^* \ to \ mathbb {C}, Z \ mapsto z^ {- 1}$ is analytical on $\ mathbb {C} ^*$ .

the function $Z \ mapsto |Z|^2 = Z \ overline {Z}$ is not analytical: it is shown that she does not admit of derived (with the direction complexes) only in 0.

the function $Z \ mapsto \ mathrm {Re} (Z)$ is not analytical: she does not admit of derived (with the direction complexes) in any point from $\ mathbb {C}$ .

(it will be noted that the two last functions admit derivative partial of all kinds)

Principal theorems on the analytical functions

The principle of the zeros isolated

One again considers an open related $U \ subset \ mathbb {C}$ and an analytical function $f: U \ to \ mathbb {C} \,$ . If $F \,$ is not the null function, then all its zero are insulated, i.e. if $a \ in U \,$ is such as $F (a)=0$ , then it exists a disc centered in $a \,$ , included in $\ U$ , such as $F \,$ does not cancel in any other point but $a \,$ on this disc; this results in:

$\ exists R >0, D (has, R) \ subset U$ and $\ forall Z \ in D (has, R) \ setminus \ {has \}, F (Z) \ neq 0$

Thus, any function

f: U \ to \ mathbb {C} \,

not constant ( IE.

\ exists (\ alpha, \ beta) \ in U^2 \ \ alpha \ neq \ beta \ and \ F (\ alpha) \ neq F (\ beta)

), is not constant in any point ( IE. in any direction starting from this point), which results in:

$\ forall has \ in U \ \ exists R >0, D (has, R) \ subset U$ and $\ forall Z \ in D (has, R) \ setminus \ {has \}, F (Z) \ neq F (a)$

One from of deduced that no analytical function

f: U \ to \ mathbb {C} \,

not constant cannot have its image

f (U) \,

contained in a real vector space of dimension 1 (in particular,

f (U) \ not \ subset \ mathbb {R}

). Indeed, like

f \,

is continuous because analytical, there should be existence of level line, and the result above the interdict, from where this last corollary.

The principle of the analytical prolongation

Lists of mathematicians having worked on the theory of the analytical functions

Ludovico Ferrari which invents into 1540 the complex numbers to solve the equation of the 4th degree.
De Moivre with which one owes the famous formulas of Moivre.
Stirling which gave celebrates it Formule of Stirling.
Euler which cleared up the theory of the logarithm. Author of the famous relation between E, I and pi. One of the first which is interested in the complex functions.
Legendre with which one owes the theory of the elliptic functions
Gauss
Laplace which invented the method of estimate of the integrals which bears its name. And which used the transformation of Laplace…
Poisson which in an article of 1813 fact the bond between " étrangetés" in real variable and behavior of the function in the complex plan.
Argand,… which interprets between 1785 and the 1830 complex numbers in geometrical term.
Cauchy and theory of residues, integral complex, ray of convergence,…
Riemann with which one owes the theorem of the application in conformity (which will be really shown only by Koebe in 1913), the theory of the function zeta of Riemann (outline),…
Weierstrass study of the singularities essential…
Casorati
Laurent development in the vicinity of a pole…
Picard its two theorems (the small one and the large one) on the exceptional values
Borel the theory of the divergent series, of the integral transformations, of the growth,…
Hadamard Theorem of decomposition, the theorem of the prime numbers,…
Pram
Jensen Formula of Jensen
Koebe
Cahen Theory of the series of Dirichlet
Montel normal families, theorem of Montel,…
Valiron
Blumenthal theory of the functions of an infinite nature

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