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 ($$\alpha \in (0;1)$$)

$f(x)$	$$D^\alpha_{a+,x}f(x)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dx}\int_a^x{\frac{f(s)ds}{(x-s)^{\alpha}}}$$
$$0$$	$$0$$
$$A$$	$$\frac{A}{\Gamma(1 - \alpha)}(x-a)^{-\alpha}$$
$$x^n$$	$$\frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)}x^{n-\alpha} + \left[ \frac{1}{\Gamma(1-\alpha)}\left(\frac{a}{x}\right)^{n+1}\left(1-\frac{a}{x}\right)^{-\alpha} - \frac{(n+1-\alpha)}{\Gamma(1-\alpha)}B_{\frac{a}{x}}(n+1, 1-\alpha) \right]x^{n-\alpha} $$
$$(x-a)^n$$	$$\frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)}(x-a)^{n-\alpha}$$
$$e^x$$	$$\frac{\gamma(1-\alpha, x-a)}{\Gamma(1-\alpha)}e^{x} + \frac{(x-a)^{-\alpha}}{\Gamma(1-\alpha)}e^a$$
$$e^{-(x-a)^2}$$	$$- (2^{\alpha -1}) \sqrt{\pi} (x-a)^{-\alpha} \left[ (x-a)^2 \ _2\tilde{F}_2 \left( \frac{3}{2},2;2-\frac{\alpha}{2},\frac{5-\alpha}{2};-(a-x)^2 \right) + (\alpha-1) \ _2\tilde{F}_2 \left( \frac{1}{2},1 ; 1-\frac{\alpha}{2},\frac{3-\alpha }{2};-(a-x)^2 \right) \right] $$
	$$D^\alpha_{b-,x}f(x)=\frac{-1}{\Gamma(1-\alpha)}\frac{d}{dx}\int_x^b{\frac{f(s)ds}{(s-x)^{\alpha}}}$$
$$0$$	$$0$$
$$A$$	$$\frac{A}{\Gamma(1 - \alpha)}(b-x)^{-\alpha}$$
$$x^n$$	$$\frac{\Gamma(\alpha-n)}{\Gamma(-n)}x^{n-\alpha} + \left[ \frac{1}{\Gamma(1-\alpha)}\left(\frac{b}{x}\right)^{n+1}\left(\frac{b}{x}-1\right)^{-\alpha} + \frac{(n+1-\alpha)}{\Gamma(1-\alpha)}B_{\frac{x}{b}}(\alpha-n-1, 1-\alpha) \right]x^{n-\alpha} $$
$$(b-x)^n$$	$$\frac{\Gamma(n+1)}{\Gamma(n+1-\alpha)}(b-x)^{n-\alpha}$$

$$\frac{(x-a)^{\alpha}}{\Gamma(\alpha + 1)}$$	$$1$$ $\alpha \in (0;1)$
$$\frac{\Gamma(n + 1)}{\Gamma(n + \alpha + 1)}(x-a)^{n+\alpha}$$	$$(x-a)^{n}$$ $\alpha \in (0;1)$
$$A(x-a)^\beta$$	$$\frac{A \cdot \Gamma(\beta + 1)}{\Gamma(\beta + 1 - \alpha)}(x-a)^{\beta - \alpha}$$
$$(x-a)^{\beta}ln(x-a)$$	$$\frac{\Gamma(\beta + 1)}{\Gamma(\beta + 1 - \alpha)}(x-a)^{\beta - \alpha} \left[ ln(x-a) + \psi^{(0)}(\beta + 1) - \psi^{(0)}(\beta + 1 - \alpha) \right]$$