418
правок
Изменения
Нет описания правки
=== Теорема о почленном дифференцировании степенного ряда ===
=== Экспонента, синус, косинус. Свойства. ===
<tex> \mathrm{exp}(0) = 1 </tex>
<tex> \mathrm{exp}(\overline{z}) = \overline{\mathrm{exp}(z)} </tex>
<tex> (\mathrm{exp}(z))' = \mathrm{exp}(z) </tex>
<tex> \mathrm{exp}(z + w) = \mathrm{exp}(z) ⋅ \mathrm{exp}(w) </tex>
<tex> \mathrm{exp}(z) ≠ 0, \ \forall z \in \mathbb{C} </tex>
<tex> \sin x = \frac{\mathrm{exp}(ix) - \mathrm{exp}(-ix)}{2i} </tex>
<tex> \cos x = \frac{\mathrm{exp}(ix) + \mathrm{exp}(-ix)}{2} </tex>
<tex> \overline{\mathrm{exp}(iz)} = \mathrm{exp}(\overline{iz}) = \mathrm{exp}(-i\overline{z}) </tex>
<tex> \cos(z) = \sum_{n=0}^{+\infty} (-1)^n \frac{z^{2n}}{(2n)!} </tex>
<tex> \sin(z) = \sum_{n=0}^{+\infty} (-1)^n \frac{z^{2n - 1}}{(2n - 1)!} </tex>
Пусть <tex> T(x) = \mathrm{exp}(ix) </tex>
<tex> T(x+y) = T(x)T(y) </tex>
<tex> \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) </tex>
<tex> \sin(x + y) = \cos(x)\sin(y) + \cos(y)\sin(x) </tex>
<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^x = 1 + x + \frac{x^2}{2} + ... </tex>
<tex> \sin(x) = x + \frac{x^3}{3} + ... </tex>
<tex> \cos(x) = 1 - \frac{x^2}{2} + ...</tex>
<tex> |x| < 1: \ (1 + x)^\alpha = 1 + \alpha x + \frac{\alpha (\alpha - 1)}{2} x^2 + ... </tex>
<tex> |x| < 1: \ \frac{1}{1-x} = 1 + x + x^2 + ... </tex>
<tex> |x| < 1: \ \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ... </tex>
=== Единственность производной ===
{{Теорема
