THE NUMBER OF ZERO SOLUTIONS FOR COMPLEX CANONICAL DIFFERENTIAL EQUATION OF SECOND ORDER WITH CONSTANT COEFFICIENTS IN THE FIRST QUADRANT

The study of complex differential equations in recent years has opened up some of questions concerning the determination of the frequency of zero solutions, the distribution of zero, oscillation of the solution, asymptotic behavior, rank growth and so on. Besides, this is solved by only some classes of differential equations. In this paper, our aim was to determine the number of zeros and their arrangement in the first quadrant, for the complex canonical differential equation of the second order. The accuracy of our results, we illustrate with two examples.

Then are generally treated canonical complex differential equations of second order with a coefficient which is an entire function.All of these studies have focused mainly on two general issues.
The first one involved the determination of the frequency of zero solutions, while the other studied the distribution and the asymptotic behavior of zero solutions in the first quadrant i.e., in the sector 0, 2 zR      .About the problem of distribution of zero solutions of complex differential equations, the case where the coefficient   Pz is quite clear.When   az is the transcendent function the situation is much more complex.Review of the scientific literature, such as (Gundersen, 1986), (Laine, 1993), (Shu Pei, 1994) and others, shows that there are mostly treated complex differential equations with transcendental coefficients z e and coefficients derived from it:   Unlike classical Nevanlinna theory, we are using the the idea of (Dimitrovski & Mijatović, 1998), (Lekić et. al., 2012), (Vujaković et al., 2011), (Vujaković, 2012) developed a new approach in determining the location and number of zero solutions.This method looked better in the applications for us.
In this paper, the subject of our considerations is complex canonical differential equations of second order with constant coefficients.

PRELIMINARIES
For complex canonical differential equation of the second order :

 
, xy  are harmonic functions, by series-iterations method which are described in detail in the works (Dimitrovski & Mijatović, 1998), (Lekić et al., 2012), (Vujaković, 2012), we get two fundamental solutions:   2 wz we called oscillatory complex functions with base   az.Mark them with (2) and (3), respectively.Further, let denotes a function of the frequency.We have seen in the papers (Vujaković et al., 2011;2016), (Vujaković, 2012) that the zero solutions (2) and ( 3) are approximately in the solutions of equations This equation is easy to solve only for the constant Zero of cosine solutions for the canonical complex differential equations of the second order (1) can be found in a similar manner.Namely, from the first equation of the system (5) we have From here, similarly as moment ago, we obtain a system of equations From here we have , that is we obtain the connection between the  and  , which does not depend on the argument 2 By the same reasoning from the system (6) we prove the assertion for

MAIN RESULTS
For starters, it is important to determine the number of zeros solutions of complex second order differential equation (1) for a constant   12 a z c ic  , or to try to evaluate zeros better.Because of the multifaceted analytic functions it is best first to observe the characteristic examples of differential equations with constant coefficients The problem is that we must know in advance for which   Because we want that z x iy  be in the first quadrant, therefore that 0, 0 xy  holds, it must be  respectively, after the well-known trigonometric identities for the sine and cosine, we get the coordinates for the zeros of sine solutions of the complex canonical differential equations of the second order: for 0,1, 2,.. n  .
From equations (12) it is evident that 0 y  , and comparing equations ( 11) with ( 12) we see that xy  .Dividing the ( 12) with ( 11) we obtain the We still need to determine how many zeros have a sine i.e., cosine solution of canonical complex differential equations of second order in the sector , 0 arg 2 Тheorem 4. In canonical complex second-order differential equation ( 9), with constant coefficients, the number of zeros of sine solutions is determined by the formula Notice that the sine zeros are equidistant because for two consecutive n and 1 n  is valid Concerning the property that the sine function within a period 2 has three zeros 0 , .....This is because in Nevanlinna theory as a measure of transcendence and infinite growth, takes the function infinity of transcendent type.