Let's write a global velocity vector (GVV) in this form \[\mathbf{V} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n, \quad \beta = v/c \qquad (2.1)\] and, proindeksirovat his time, we find the global vector length (GVL): \[\mathbf{L} = c \int \frac{1}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n\, \Bbb{d}t \qquad (2.2)\] With this we mean that in the General case, the speed \(v\) depends on time, hence: \(\beta = \beta(t)\). Now take the global vector force (GVF) from here and rewrite it in this form: \[\mathbf{F} = m_0 c \beta^{'} \sum \limits_{n=0}^{\infty} \mathbf{j_n} n\beta^{(n-1)} \qquad (2.3)\] where: \(\beta^{'} = \Bbb{d}\beta / \Bbb{d}t\). In order to find work (energy) according to classical concepts, it is necessary to integrate a force increment of the path: \[E = \int \mathbf{F} \,\Bbb{d} \mathbf{L} \qquad (2.4)\] mediterrannee Transforming the expression we get: \[E = \int \mathbf{F}\cdot \mathbf{V} \,\Bbb{d}t \qquad (2.5)\] Next, we substitute back the formula (2.1) and (2.3): \[E = m_0 c^2 \int \frac{\beta^{'}}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} n\beta^{(n-1)} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n \,\Bbb{d}t \qquad (2.6)\] Scalar multiply these sums in expanded terms, we get: \[E = m_0 c^2 \int \frac{\beta}{\gamma} \sum \limits_{n=0}^{\infty} n \beta^{2(n-1)} \,\Bbb{d}\beta \qquad (2.7)\] the Sum of the power series is: \[\sum \limits_{n=0}^{\infty} n x^{2(n-1)} = {1 \over (1 - x^2)^2} \qquad (2.8)\] Making the substitution in expanded terms, and substituting back the sum of the series, we get the final integral: \[E = {m_0 c^2 \over 2} \int {\Bbb{d}\beta^2 \over (1 - \beta^2)^{3/2}} \qquad (2.9)\] Taking this integral, we find that: \[E = m_0 c^2 \gamma = m c^2 \qquad (2.10)\] This formula is fully consistent with obtained in this section.

Collect the formulas into a single number: \[E = \mathbf{P}\cdot \mathbf{V} = \int \mathbf{V} \,\Bbb{d}\mathbf{P} = \int \mathbf{F} \,\Bbb{d} \mathbf{L} = m c^2 \qquad (2.11)\] we only Recall that the global vector of the momentum \(\mathbf{P}\), and related formulas we took out.