tag:
разложение_функции_в_непрерывную_дробь
§
$$\operatorname{arctg} x = \cfrac{x}{{1 + \cfrac{{{x^2}}}{{3 + \cfrac{{4{x^2}}}{{5 + \cfrac{{9{x^2}}}{{7 + \cfrac{{16{x^2}}}{{...}}}}}}}}}}$$§
\[\left[ {{\text{Ламберт}}} \right]\]
$$\operatorname{tg} x = \cfrac{x}{{1 - \cfrac{{{x^2}}}{{3 - \cfrac{{{x^2}}}{{5 - \cfrac{{{x^2}}}{{7 - ...}}}}}}}} = \cfrac{1}{{\cfrac{1}{x} - \cfrac{1}{{\cfrac{3}{x} - \cfrac{1}{{\cfrac{5}{x} - \cfrac{1}{{\cfrac{7}{x} - \cfrac{1}{{...}}}}}}}}}}$$1722.
\[\tanh \frac{1}{n} = \frac{1}{{n + \frac{1}{{3n + \frac{1}{{5n + \frac{1}{{7n + \frac{1}{{...}}}}}}}}}}\]
1978.
\[\coth x = \frac{{{e^{2x}} + 1}}{{{e^{2x}} - 1}} = \frac{1}{x} + \frac{{\frac{1}{{1 \cdot 3}}}}{{\frac{1}{x} + \frac{{\frac{1}{{3 \cdot 5}}}}{{\frac{1}{x} + \frac{{\frac{1}{{5 \cdot 7}}}}{{\frac{1}{x} + ...}}}}}}.\]
2154.
$$\eqalign{
& s > 0,{\text{ }}s \ne \frac{1}{n}; \cr
& \cfrac{1}{{\cfrac{s}{{1 + \cfrac{{s - 1}}{{1 + \cfrac{{2s}}{{1 + \cfrac{{2s - 1}}{{1 + \cfrac{{3s}}{{1 + \cfrac{{3s - 1}}{{1 + ...}}}}}}}}}}}}}} = {s^{\frac{1}{s} - 1}} \cdot {e^{\frac{1}{s}}} \cdot \Gamma \left( {\frac{1}{s},\frac{1}{s}} \right) \cr} $$