Tables with integrals and differentials

Таблицы

Таблица 1 - m-кратный дифференциала

$№$	$$f(x)$$	$$\frac{d^m}{dx^m}f(x)$$
1.	$$x^n$$	$$\frac{\Gamma(n+1)}{\Gamma(n-m+1)}x^{n-m}$$
2.	$$f(x)g(x)$$	$$\sum_{i=0}^{m}{ {m \choose i} \frac{d^{m-i}}{dx^{m-i}}f(x) \frac{d^i}{dx^i}g(x) }$$

Таблица 2 - дробные производные

$№$	$f(x)$	$$D^\alpha_{a+,x}f(x)=\frac{1}{\Gamma(m-\alpha)}\frac{d^m}{dx^m}\int_a^x{\frac{f(s)ds}{(x-s)^{1+\alpha-m}}}$$	$$D^\alpha_{b-,x}f(x)=\frac{(-1)^m}{\Gamma(m-\alpha)}\frac{d^m}{dx^m}\int_x^b{\frac{f(s)ds}{(s-x)^{1+\alpha-m}}}$$
0.	$$0$$	$$0$$	$$0$$
1.	$$A$$	$$\frac{A}{\Gamma(1 - \alpha)}(x-a)^{-\alpha}$$	$$\frac{A}{\Gamma(1 - \alpha)}(b-x)^{-\alpha}$$
2.	$$C_1(x-a)^{\alpha - 1}$$	$$0$$ $\alpha \in (0;1)$
3.	$$\frac{(x-a)^{\alpha}}{\Gamma(\alpha + 1)}$$	$$1$$ $\alpha \in (0;1)$
4.	$$\frac{\Gamma(n + 1)}{\Gamma(n + \alpha + 1)}(x-a)^{n+\alpha}$$	$$(x-a)^{n}$$ $\alpha \in (0;1)$
5.	$$A(x-a)^\beta$$	$$\frac{A \cdot \Gamma(\beta + 1)}{\Gamma(\beta + 1 - \alpha)}(x-a)^{\beta - \alpha}$$
6.	$$(x-a)^{\beta}ln(x-a)$$	$$\frac{\Gamma(\beta + 1)}{\Gamma(\beta + 1 - \alpha)}(x-a)^{\beta - \alpha} \left[ ln(x-a) + \psi^{(0)}(\beta + 1) - \psi^{(0)}(\beta + 1 - \alpha) \right]$$