## 6 Propagation Time for a Signal, Emitted and Intercepted by Satellites on a Space-Oriented Orbit
### Another Integral in the Weierstrass Form After Applying the Four-Dimensional Uniformization  
Further, by \"four-dimensional uniformization\", it shall be meant that the polynomial of the fourth degree under the square root in the denominator of the elliptic integral shall be \"uniformized\" by introducing complicated functions, depending on the Weierstrass function and its derivative, which represent functions of the complex variable \(z\). The term \"uniformization\" shall be clarified in the next section. The Standard Method for \(3d\) Uniformization in Elliptic Functions Theory--Reminder of the Basic Facts  
In accord with the standard notions in elliptic functions theory [34; 55], the s.c. \"fundamental parallelogram\" is defined on the two-periodic lattice (see also the monograph [56])  
\[\Lambda\equiv\bigg{\{}m\omega_{1}+n\omega_{2}\ ;0\leq m\leq 1\,0\leq n\leq 1, Im\bigg{(}\frac{\omega_{1}}{\omega_{2}}\bigg{)}>0\bigg{\}}. \tag{177}\]  
The two periods \(\omega_{1}\) and \(\omega_{2}\) constitute the basis on the complex plane and, thus, each complex number can be represented as \(z=m\omega_{1}+n\omega_{2}\). Then, the Weierstrass function  
\[\rho(z)\equiv\frac{1}{z^{2}}+\sum\bigg{(}\frac{1}{(z-\omega)^{2}}-\frac{1}{ \omega^{2}}\bigg{)} \tag{178}\]  
is an example of an elliptic function. By definition, \(f(z)\) is an elliptic function if it is meromorphic and doubly periodic. The double periodicity of the elliptic function means that the equality  
\[f(z+n\omega_{1}+m\omega_{2})=f(z) \tag{179}\]  
is fulfilled, where \(n\) and \(m\) are integer numbers. A complex function is meromorphic, if it does not have singular points other than poles [34]. The singular points can be of three kinds [57; 58]:  
1. Removable singular points, for which \(\lim_{z\to a}f(z)\) exists and is a finite one. These points constitute the s.c. main part in the Laurent decomposition \[f(z)=c_{0}+c_{1}(z-a)+......+c_{n}(z-a)^{n}\Rightarrow\lim_{z\to a}f(z)=c_{0}.\] (180)
2. Essentially singular points, for which \(\lim_{z\to a}f(z)\) does not exist. 3. The case with poles, for which \(\lim_{z\to a}f(z)=\infty\). For such a case, the Laurent decomposition is \[f(z)=\frac{c_{-n}}{(z-a)^{n}}+........+\frac{c_{-1}}{z-a}+\sum_{k=0}^{\infty}c _{k}(z-a)^{k}.\] (181)The first derivative of the Weierstrass function \(\rho(z)\) (178) is  
\[\rho^{\prime}(z)\equiv-\frac{2}{z^{3}}-2\sum\frac{1}{(z-\omega)^{3}} \tag{182}\]  
and it satisfies the parametrizable form of the cubic algebraic equation  
\[y^{2}=4x^{3}-g_{2}x-g_{3}\ \text{i.e.,}\ x\equiv\rho(z),\ \ y\equiv\rho^{\prime}(z), \tag{183}\]  
where the functions \(\rho(z)\) and \(\rho^{\prime}(z)\) are called also uniformization functions for the above cubic algebraic curve \(y^{2}=4x^{3}-g_{2}x-g_{3}\). The summation in the defining formulaes (178) for the Weierstrass function \(\rho(z)\) is over all points on the two-dimensional lattice on the complex plane, i.e., on all possible numbers \(m\) and \(n\). The coefficient functions \(g_{2}\) and \(g_{3}\) are the s.c. \"invariants of the Weierstrass function\" \(\rho(z)\)  
\[g_{2}=60G_{4}=60\sum\frac{{}^{\prime}1}{\omega^{4}}\ \,\ g_{3}=140G_{6}=140 \sum\frac{{}^{\prime}1}{\omega^{6}}, \tag{184}\]  
defined as infinite convergent sums over the period \(\omega\) of the two-periodic lattice. The prime \({}^{\prime\prime\prime}\) above the two sums means that the period \(\omega=0\) is excluded from the summation, so that the expressions would not tend to infinity. Application of the Weierstrass Integral and of the Weierstrass Elliptic Curve in the Parametrizable Form  
This section has the purpose of clarifying the importance of finding the Weierstrass invariants \(g_{2}\) and \(g_{3}\) for the calculation of an elliptic integral in the Weierstrass form \(\int\frac{dx}{\sqrt{4x^{3}-g_{2}x-g_{3}}}\).