Holomorphic function

The holomorphic functions constitute the central pillar of the study of the Analyze complexes; they are the functions with values in $\ mathbb C$ , definite and derivable in any point of an open subset of the plane complex $\ mathbb C$ .

This condition is much stronger than the real derivability. It implies (via the theory of Cauchy) that the function is analytical: it is indefinitely derivable and is equal in the vicinity of any point of opened with the sum of its Série of Taylor. A remarkable fact results from this: the concepts of analytical function complex and holomorphic function coincide.

Definition

Are U an open subset (not vacuum) of the unit $\ mathbb C$ of the complex numbers and a function $f: U \ to \ mathbb C$ .

One says that F is derivable (with the direction complexes) in a point Z ₀ of U if the limit following, called derived from F in Z ₀ exists:

$f' (z_0) = \ lim_ {Z \ to z_0} {F (Z) - F (z_0) \ over Z - z_0}$

The limit is taken here on all the continuations of complex numbers tending towards Z ₀, and for all these continuations the quotient must tend towards the same number F '( Z ₀). Intuitively, if F is derivable with the complex direction in Z ₀, and if one approaches the point Z ₀ in the direction of a vector U , then (provided that F “( Z ₀) ≠ the 0) images will approach the point F ( Z ₀) in the direction of the vector F ” ( Z ₀) U (produced complex numbers F '( Z ₀) and U ).

The rules of calculation of derived with the direction complexes are identical to those of the Dérivée S from the functions of a real variable: linearity, derived from a product, a quotient, a made up function.

One says that F is holomorphic on open U if it is derivable (with the direction complexes) in all not Z ₀ of U . In particular, one calls whole Fonction a holomorphic function in all the complex plan.

Examples

All polynomial function with Coefficient S complexes is holomorphic on $\ mathbb C$ .
All rational Fonction with complex coefficients is holomorphic on the complementary one to the whole of its poles. For example, the function reverses $z \ mapsto 1/z$ is holomorphic on $\ mathbb C^*$ .
Is $\ sum_ {N \ geq0} a_n z^n$ a whole series with complex coefficients of ray of convergence not no one (finished or not); one notes $D$ his disc of convergence.

the function

f: D \ to \ mathbb C

defined by

f (Z) = \ sum_ {n=0} ^ {+ \ infty} a_n z^n

is holomorphic, and for all

z \ in D

f' (Z) = \ sum_ {n=1} ^ {+ \ infty} N a_n z^ {n-1}

makes some, this function is indefinitely derivable on

D

the exponential function is holomorphic on $\ mathbb C$ . It is the same of the goniometrical functions (which can be defined starting from the exponential function by means of the formulas of Euler) and of the hyperbolic functions.
One calls determination of the logarithm on open a $\ U$ of $\ mathbb C^*$ any holomorphic function $L: U \ to \ mathbb {C}$ such as $\ forall \, Z \ in U, \, \ exp (L (Z)) = z$ . One has then: $\ forall \, Z \ in U, \, L \, “(Z) = 1 \,/\, z$ .

On all open $U$ of $\ mathbb C^*$ where $L$ logarithm exists a determination, one can define, whatever the $k \ in \ mathbb {Z}$ , the function $L_k: U \ to \ mathbb {C}, \, Z \ mapsto L (Z) + 2 \, K \, \ pi \, i$ . Each one of these functions is a determination of the logarithm on $U$ , and reciprocally, if $U$ is related, any determination of the logarithm on $U$ is one of these functions.
the existence on open related a $U$ of $\ mathbb C^*$ of a determination of the logarithm is equivalent to the existence of a holomorphic function $\ ell$ on this opened, such as $\ ell” (Z) = \ frac {1} {Z}$ for all $z \ in U$ ; in this case, there exists (at least) constant a complex $C$ such as the function $L: U \ to \ mathbb {C}, Z \ mapsto \ to elect (Z) + C$ is a determination of the logarithm on $U$ .

There does not exist determination of the logarithm on open the $\ mathbb C^*$ .
There exists a determination of the logarithm on the whole of the complex numbers private of the half-line of negative or null realities (one speaks about " cut "). Among all the determinations of the logarithm on this opened, there is one and only one which prolongs the real Napierian logarithm.
more generally, there exists a determination of the logarithm on all open Simplement related not containing 0.

On all open $U$ of $\ mathbb C^*$ where $L$ logarithm exists a determination, one can define a determination of the power of exhibitor $\ a$ (where $a \ in \ mathbb {C}$ ), by posing

\ forall \, Z \ in U, \, z^a = \ exp (\ has, L (Z))

In particular, if

n

is a natural entirety equal to or higher than 2, the function

U \ to \ mathbb {C}, \, Z \ mapsto z^ {\ frac {1} {N}} = \ exp \ left (\ frac {1} {N} \, L (Z) \ right)

is holomorphic on

U

and checks the identity

\ forall \, Z \ in U, \, \ left (z^ {\ frac {1} {N}} \ right) ^n = z

One says that this function is a determination on

U

of the root N - ième.

One can note

\ \ sqrt {Z}

instead of

\ z^ {\ frac {1} {N}}

(so of strictly positive realities belong to

U

, it may be that there is then conflict between this notation, and its usual significance, being used to indicate the root N - ième positive).

the reciprocal goniometrical functions have same manner of the cuts and are holomorphic safe everywhere with the cuts.

Properties

Because complex derivation is linear and that she obeys the traditional rules of derivation, the sums, products or made up of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic on all open where the denominator is not cancelled.

If one identifies $\ mathbb C$ with $\ mathbb R^2$ , then the holomorphic functions on open of $\ mathbb C$ coincide with the functions of two real variables which are $\ R$ -différentiables on this opened and check there the equations of Cauchy-Riemann, a system of two partial derivative equations.

Close to a point $z_0$ where its derivative is nonnull, a holomorphic function $f$ is a Transformation in conformity, i.e. it preserves the angles (directed) and the forms of small figures (but not lengths, in general). Indeed, its differential at the point $z_0$ is the application $\ mathbb {C}$ -linéaire $df_ {z_0}: \ mathbb {C} \ to \ mathbb {C}, \, U \ mapsto has \, u$ , where $A = f' (z_0) \ neq 0$ : the differential is thus identified with a direct Similitude of the plan.

It is established (by means of the integral Formule of Cauchy) that any holomorphic function on open a U is indefinitely derivable in any point compared to the complex variable. Such a function coincides in the vicinity of any point $z_0$ of U with its Série of Taylor in this point (it is analytical), and the series converges on any open disc of center $z_0$ and included in U . The Taylor series can converge on a larger disc; for example, the Taylor series of the logarithm converges on any disc not containing 0, even in a vicinity of the strictly negative real numbers.

Integral formula of Cauchy, one deduces in particular that any holomorphic function on open containing a closed disc is completely given inside this disc by its values on the border of this one.

See too

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