Изменения

1.1) <tex> \mathrm{exp}(0) = 1 </tex>

1.2) <tex> \mathrm{exp}(\overline{z}) = \overline{\mathrm{exp}(z)} ; \ /S_n(\overline{z}) = \overline{S_n(x)})/</tex>

1.3) <tex> (\mathrm{exp}(z))' = \mathrm{exp}(z) ; \ /\sum_{n = 1}^{+ \infty} (\frac{z^n}{n!})' = \sum_{n = 1}^{+ \infty} \frac{z^{n - 1}}{(n - 1)!} = \sum_{n = 0}^{+ \infty} \frac{z^n}{n!}/ </tex>

1.4) <tex> (\mathrm{exp}(z + wx) = \mathrm{exp}(z) ⋅ \mathrm'|_{expx = 0}(w) = 1 </tex>

{{Теорема|statement=<tex> \forall z, w \in \mathbb{C} : \mathrm{exp}(z+ w) = \mathrm{exp}(z) ⋅ \mathrm{exp}(w) ≠ 0, </tex>|proof=<tex> \ sum \forall frac{z ^n}{n!} \in cdot \sum \mathbbfrac{w^k}{Ck!} </tex>

<tex> \sin x sum_{k = 0}^{+ \infty} \frac{(z + w)^k}{k!} = \mathrmsum_{expk = 0}^{+ \infty} \sum_{l = 0}^{k} \frac{z^l}{l!} \cdot \frac{w^{k - l}}{(ixk - l) - !} = \sum_{l = 0}^{+ \infty} \sum_{k = l}^{+ \infty} \mathrmfrac{expz^l}{l!} \cdot \frac{w^{k - l}}{(k -ixl)!}{2i} = </tex>

<tex> = \cos x sum_{l = 0}^{+ \infty} \sum_{n = 0}^{+ \infty} \frac{z^l}{l!} \cdot \frac{w^n}{n!} = \mathrmsum_{l = 0}^{exp+ \infty}(ix) \frac{z^l}{l!} \cdot \sum_{n = 0}^{+ \mathrminfty} \frac{w^n}{expn!}) = (-ix\sum \frac{w^n}{n!})(\sum \frac{z^l}{2l!} ) </tex>}}

* Следствие: <tex> \overline{\mathrm{exp}(izz)} = \mathrm{exp}(\overline{iz}) = \mathrm{exp}(-i\overline{ne 0 </tex> — ни при каких <tex> z}) </tex>

2.1) <tex> \cos(z) sin x = \sum_frac{n=0}^\mathrm{+\inftyexp} (ix) -1)^n \fracmathrm{z^{2nexp}(-ix)}{(2n)!2i} </tex>

2.2) <tex> \sin(z) cos x = \sum_frac{n=0}^\mathrm{+\inftyexp} (-1ix)^n + \fracmathrm{z^{2n - 1exp}}{(2n - 1ix)!}{2} </tex>

2.3) <tex> \cos(z) = \sum_{n=0}^{+\infty} (-1)^n \frac{z^{2n}}{(2n)!} </tex> 2.4) <tex> \sin(z) = \sum_{n=0}^{+\infty} (-1)^n \frac{z^{2n - 1}}{(2n - 1)!} </tex> 2.5) Пусть <tex> T(x) = \mathrm{exp}(ix) </tex>

2.6) <tex> |T(x)| = 1 </tex> <tex> ; \ \cos^2(x) + \sin^2(x) = 1 </tex> <tex> \lim_{x \to 0} \frac{1-cos(x)}{x^2} = \frac{1}{2} </tex> <tex> \lim_{x \to 0} \frac{1-cos(x)}{x} = 0 </tex>

<tex> e^(\frac{T(x = 1 ) + T(-x )}{2})^2 + (\frac{T(x) - T(-x)}{2i})^2}= T(x)T(-x) = T(0) = \mathrm{2exp} + ... (i0) = 1 </tex>

2.7) <tex> \lim_{x \to 0} \frac{\sin(x) }{x} = 1; \ \lim_{x + \to 0} \frac{1 - \cos(x)}{x^32} = \frac{1}{32} + ... </tex>

<tex> \lim_{x \to 0} (\frac{\mathrm{exp}(ix) - 1}{ix}) = \lim_{x \to 0} (\frac{\cos(x) = - 1 - }{ix} + \frac{i \sin(x^2)}{2ix} + ...) </tex>

-----<tex> |x| < 1: \ (in \mathbb{C} \begin{cases} e^x = 1 + x)+ \frac{x^2}{2} + \ldots \\ \alpha sin(x) = 1 x + \alpha frac{x ^3}{3} + \frac{ldots \\ \alpha cos(x) = 1 - \alpha - 1)frac{x^2}{2} x^2 + ... \ldots \end{cases} </tex>

<tex> |x| < 1: \begin{cases} (1 + x)^\alpha = 1 + \alpha x + \frac{\alpha (\alpha - 1)}{2} x^2 + \ldots \\ \frac{1}{1-x} = 1 + x + x^2 + ... \ldots \\ \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots \end{cases}</tex>

<tex> |x| \sum a_k \to </tex> Абель < 1: tex> \to \ sum a_k \lncdot x^k = f(1 + x) = x - ; \fraclim_{x^2\to 1- 0}{2} + \frac{f(x^3}{3} - ... ) = S </tex>