# Algorithms for Calculating Generalized Trigonometric Functions
## 3 Definitions and Properties of _Ateb_-Functions  
In this section, we show the nonlinear first-order differential equation system solution based on _Ateb_-functions. The results presented in this section are based mainly on references [3; 8; 9]. The concept of _Ateb_-functions facilitates the analytical solution of differential equation systems that characterize highly nonlinear processes in a medium with one degree of freedom  
\[ \begin{cases}\dot{x}+\beta y^{m}&=0,\\ \dot{y}+\alpha x^{n}&=0,\end{cases}  \tag{1}\]where \(\alpha\) and \(\beta\) are some real constants, and  
\[n=\frac{2\theta_{1}^{\prime}+1}{2\theta_{1}^{\prime\prime}+1},\;m=\frac{2\theta_ {2}^{\prime}+1}{2\theta_{2}^{\prime\prime}+1},\;(\theta_{1}^{\prime},\theta_{1} ^{\prime\prime},\theta_{2}^{\prime},\theta_{2}^{\prime\prime}=0,1,2,...). \tag{2}\]  
_Ateb_-functions are mathematically defined through the inversion of the incomplete _Beta_-function. This approach not only defines these functions but also inspired their name, as _Ateb_ is derived from inversing the term _Beta_. An incomplete _Beta_-function is defined by the next formula  
\[B_{x}(p,q)=\int_{0}^{x}t^{p-1}(1-t)^{q-1}dt, \tag{3}\]  
where \(p\) and \(q\) are real numbers. In the special case where \(x=1\), Equation (3) is simplified to the first-kind Euler integral:  
\[B_{1}(p,q)=\int_{0}^{1}t^{p-1}(1-t)^{q-1}dt, \tag{4}\]  
which represents the _Beta_-function, denoted as \(B(p,q)\). For each \(x\) from the interval [0, 1], functions \(B_{x}(p,q)\) and \(B_{1}(p,q)\), defined by expressions (3) and (4), are non-negative and satisfy the following properties:  
\[0\leq B_{x}(p,q)\leq B_{1}(p,q),\] \[B_{x}(p,q)=B_{1}(p,q)-B_{1-x}(p,q).\]  
Let us discuss two specific cases:  
\[p=\frac{1}{n+1},\;q=\frac{1}{m+1}; \tag{5}\] \[p=\frac{1}{n+1},\;q=\frac{m}{m+1}-\frac{1}{n-1}, \tag{6}\]  
where \(m\) and \(n\) are determined by Formula (2). When \(p>0\) and \(q>0\), then the _Beta_-function is well defined and determined. For other real values of \(p\) and \(q\), the Beta function heads to infinity at \(t\to 0\) or at \(t\to 1\). _Ateb_-functions for values (5) are named periodical, and those for values (6) are named hyperbolic or aperiodic _Ateb_-functions. System (1), if parameters \(m\), \(n\) satisfy Formula (5), describes oscillatory motion, and, if \(m\), \(n\) satisfy Formula (6), it describes hyperbolic or aperiodic motion. If \(m=1\) and \(n\) is defined by (2), then system (1) can be rewritten as:  
\[\ddot{x}+c|x|\cdot x^{\theta-1}=0, \tag{7}\]  
where \(\theta\) depends on the parameters \(\theta_{1}^{\prime}\) and \(\theta_{2}^{\prime}\) in Formula (2). Let us evaluate the expression  
\[\omega=\frac{1}{2}\int_{0}^{-1\leq y\leq 1}t^{-\frac{n}{n+1}}(1-t)^{-\frac{n}{m +1}}dt. \tag{8}\]  
where parameters \(m\), \(n\) are determined by (5) and (2). Let us perform the placement of variable  
\[t=v^{-n+1}. \tag{9}\]Equation (8) is transformed into the following expression  
\[B_{x}(p,q)=\int_{0}^{x}t^{p-1}(1-t)^{q-1}dt, \tag{3}\]0  
In Formula (10), \(\omega\) has a dependence from variable \(v\), and from the parameters \(m\) and \(n\). For the construction of _Ateb_-functions, we study the inverse dependence of \(v\) from \(\omega\).