Identity and extensions

1) Zero of one function analytics :

Point z₀ pertaining to the G dominion says zero of f(z)se f(z₀) = 0. From the development of f(z) in series of powers in around of point z₀ f(z) = Sc_n(z-z₀)ⁿ , follows that coefficient c₀ is equal to zero. If also the coefficients until to the k-1 are equal to zero and coefficient c_k is various from zero then point z₀ says zero of order k of the function f(z).

 

2) Identity of series of powers :

Such ones are given to 2 series of potenze e in convergent the same circle with centerz ₀, that their sums coincide in with of infinites aim z ¹ zhaving ₀ z₀ like accumulation point. Then to_n = b_n .

 

3) It is f(z) analytics in a G dominion and that cancellations in others points z_n ? G then if the succession {z_n} converges to the pertaining limit to the same dominion, then the function f(z) is identically equal to zero in the G dominion :

One before demonstrates that f(z) = the 0 to the inside of the circle |z-a| < R₀ taking advantage of repeatedly the fact that f_n(a) = 0, the result is that all the c_n are null and after all therefore the function is null. In order to demonstrate that f(z) = 0 in all the dominion instead enough to demonstrate that is worth 0 in z₁ that obtains combining with a curve to and z₁ , taking to the point of intersection between the edge of the circle of beam ₀R and the curve, a new beam of convergence in which f(z) = the 0 is found, iterando it is reached z₁ .

 

4) a function f(z) ¹ 0, analytics in a G dominion , does not have that an ended number of zeroes in every sottodominio closed limited of the G dominion:

If the number of the zeroes were infinite, from it a convergent subsuccession to a point to in could be extracted which the function is worth 0, that it denies the hypotheses.

 

5) Theorem of oneness :

If two functions f(z) and j(z) are analytics in a G dominion in which a succession of points {z _n}in which exists the values of the functions f(z) and j(z) coincides, then f(z) = j(z) in G.

He is sufficient to establish that the function y(z) = f(z) - j(z) = 0 in G.

 

6) regular Point :

A point z₀ pertaining to a limited dominion is said to regulate for the function f(z) if convergent series of powers S c _n(z-z₀ existsone)ⁿ that, in the intersection of the G dominion with its circle of convergence |z-z₀| < r(z₀), converge to the function f(z).

 

7) On the frontier of the circle of convergence of a series of powers a singular point of the function lies at least F(z) analytics, to which the given series converges :

For absurdity it is had that all the points of the edge of the circle of convergence of the series are regular that is that in the intersection between the convergence circle that corresponds to the single point and the circle of convergence of the series it begins them has convergence to f(z), are had that the difference between the beams of the relative circles of convergence you to 2 points z₁ and z₂ that find on the edge of the circle he is smaller of the distance between the two heads that it is equivalent to say that r(z) is a function uniform continuous let alone limited inferiorly (r(z) > 0) and therefore it assumes its absolute minimum on C₀ after all obtains that the convergence beam must be R₀ r₀ > R₀ and therefore contraddice the hypothesis begins them.

 

8) Function analytics total :

Draft of the F(z) function, obtained for analytical extension along all the possible chains of dominions that they exit from definition the G dominion begins them of the function analytics f(z).

 

9) Disc of centered massimale analiticità in z₀ :

Draft of a dominion that properly is not contained in some disc of center z₀ in which f is olomorfa.