Całki

1. $I_{n} = \int_{}^{}\frac{\text{dx}}{{(x^{2} + 1)}^{n}};\ \ \ I_{n} = \ \frac{x}{2(n - 1){(1 + x^{2})}^{n - 1}} + \ \frac{2n - 3}{2n - 2}\ I_{n - 1}$

2. $\int_{}^{}{\frac{\text{dx}}{\sqrt{x^{2} + k}} = \ln\left| x + \sqrt{x^{2} + k} \right| + C\ \ \ }$

3. $\int_{}^{}{\frac{\text{dx}}{\sqrt{a^{2} - x^{2}}} = \arcsin{\frac{x}{|a|} + C\ \ \ }}$

4. $\int_{}^{}{\sqrt{a^{2} - x^{2}}dx = \ \frac{a^{2}\arcsin\frac{x}{a}}{2} + \frac{\text{x\ }\sqrt{a^{2} + x^{2}}}{2} + C\ \ \ \ }$

5. $\int_{}^{}\sqrt{x^{2} + k}dx = \ \frac{1\ }{2}x\sqrt{x^{2} + k} + \ \frac{1}{2}k\ln\left| x + \ x^{2} + k \right| + C$

Całkowanie wyrażeń trygonometrycznych

Stosujemy podstawienie t = tg$\ \frac{x}{2};\ \sin{x = \ \frac{2t}{1 + t^{2}}}\ ;\cos{x = \ \frac{1 - \ t^{2}}{1\ + \ t^{2}};\ \ }$

$dx = \frac{2\ dt}{1 + \ t^{2}}$.

Lub podstawienie $t = tg\ x;\ \sin^{2}x = \ \frac{t^{2}}{1 + \ t^{2}};\ \cos^{2}x = \ \frac{1}{1 + \ t^{2}};$

$dx = \ \frac{\text{dt}}{1 + \ t^{2}};\operatorname{\ sin}{x\cos{x = \ \frac{t}{1 + \ t^{2}}}}$.

Współrzędne sferyczne:∖n

$$\left\{ \begin{matrix} x = r\cos{\theta\cos\varphi} \\ y = r\ cos\theta\sin\varphi \\ z = r\sin\theta \\ \end{matrix} \right.\ $$	$${\text{φϵ}\left\lbrack 0;2\pi \right\rbrack\backslash n}{\text{θϵ}\left\lbrack - \frac{\pi}{2};\frac{\pi}{2} \right\rbrack}$$ $$\frac{D(x,y,z)}{D(r,\varphi,\theta)} = r^{2}\cos\theta$$

Szereg trygonometryczny Fouriera

$$f\left( x \right)\sim\frac{a_{0}}{2} + \ \sum_{n = 1}^{\infty}{a_{n}\ \cos{\frac{\text{n\ π\ x}}{l} + \ b_{n}}}\sin\frac{\text{n\ π\ x}}{l}$$

$$a_{0} = \ \frac{1}{l}\ \int_{- \ l}^{l}{f\left( x \right)\text{dx}}$$

$$a_{n} = \ \frac{1}{l}\ \int_{- \ l}^{l}{f\left( x \right)\cos\frac{\text{n\ π\ x}}{l}\text{dx}}$$

$$b_{n} = \ \frac{1}{l}\ \int_{- \ l}^{l}{f\left( x \right)\sin\frac{\text{n\ π\ x}}{l}\text{dx}}$$

Postać zespolona

$$f\left( x \right)\sim\sum_{- \ \infty}^{\infty}{c_{n}\ e^{\text{i\ }\frac{\text{n\ π\ x}}{l}}}$$

$$c_{n} = \ \frac{1}{2l}\ \int_{- \ l}^{l}{f\left( x \right)\text{\ e}^{- i\ \frac{\text{n\ π\ x}}{l}}\text{dx}}\text{\ \ \ lub\ \ }c_{n} = \ \frac{1}{2l}\ \int_{0}^{2l}{f\left( x \right)\text{\ e}^{- i\ \frac{\text{n\ π\ x}}{l}}\text{dx}},\ \ gdy\ f\left( x \right)ma\ okres\ 2l$$

Całka Fouriera

f(t) =  ∫₀^∞[a(ω)cosωt + b(ω)sinωt] dω

$$a\left( \omega \right) = \ \frac{1}{\pi}\ \int_{- \infty}^{\infty}{f\left( r \right)\cos\text{ωr\ dr}}$$

$$b\left( \omega \right) = \ \frac{1}{\pi}\ \int_{- \infty}^{\infty}{f\left( r \right)\sin\text{ωr\ dr}}$$

Postać zespolona

$$f\left( t \right) = \ \frac{1}{2\pi}\ \int_{- \infty}^{\infty}{e^{\text{i\ ω\ t}}\text{\ dω}}\int_{- \infty}^{\infty}{f\left( r \right)\operatorname{}\text{\ dr}}$$

Transformata Laplace’a

$$\text{t\ f}\left( t \right) - \frac{d\overset{\overline{}}{f}(s)}{\text{ds}}$$

$$e^{- \alpha\ t}\text{\ f}\left( t \right)\ \overset{\overline{}}{f}(s + \ \alpha)\ $$

$$f\left( t - \ t_{0} \right)\text{\ \ }\text{\ \ }e^{- \ t_{0}\text{\ s}}\overset{\overline{}}{f}(s)\ \ $$

$$f\left( \text{at} \right)\ \text{\ \ }\frac{1}{a}\ \overset{\overline{}}{f}\ \left( \frac{s}{a} \right)$$

$$f^{'}\left( t \right)\ \text{\ \ s\ }\overset{\overline{}}{f}\left( s \right) - f(0)$$

$$f^{''}\left( t \right)\ \text{\ \ }s^{2}\ \overset{\overline{}}{f}\left( s \right) - sf\left( 0 \right) - f^{'}\left( 0 \right)$$

$$f^{(3)}\left( t \right)\ \text{\ \ }s^{3}\ \overset{\overline{}}{f}\left( s \right) - s^{2}f\left( 0 \right) - \text{s\ f}^{'}\left( 0 \right) - f^{''}\left( 0 \right)$$

Tabela

f(t), t > 0	$$\overset{\overline{}}{f}(s)$$
1	$$\frac{1}{s}$$
t^n	$$\frac{n!}{s^{n + 1}}$$
e^a t	$$\frac{1}{s - a}$$
cosωt	$$\frac{s}{s^{2} + \ \omega^{2}}$$
sinωt	$$\frac{\omega}{s^{2} + \ \omega^{2}}$$
$$\operatorname{t\ sin}\text{ωt}$$	$$\frac{2\ \omega\ s}{{(s^{2} + \ \omega^{2})}^{2}}$$
$$\operatorname{t\ cos}\text{ωt}$$	$$\frac{s^{2} - \ \omega^{2}}{{(s^{2} + \ \omega^{2})}^{2}}$$
sin^2ωt	$$\frac{2\ \omega^{2}}{s(s^{2} + 4\omega^{2})}$$
cos^2ωt	$$\frac{s^{2} + \ 2\ \omega^{2}}{s(s^{2} + 4\omega^{2})}$$
t^n e^a t	$$\frac{n!}{{(s - a)}^{n + 1}}$$
e^a tsinωt	$$\frac{\omega}{{(s - a)}^{2} + \ \omega^{2}}$$
e^a tcosωt	(s − a)^2
(1  at)e^−a t	$$\frac{s}{{(s + a)}^{2}}$$

Transformata Fouriera

F(iω) =  ∫_−∞^∞e^{i ω t} f(t)dt

ℱ[f(t)] = f(iω) // f(t) jest funkcją rzeczywistą

1. Liniowość

ℱ[λ₁f₁(t)+ λ₂f₂(t)] = λ₁ F[ f₁(t)] +  λ₂ ℱ[ f₂(t)] , λ₁, λ₂  ∈ R

1. Pochodna ℱ−transformaty

$$\frac{d^{k}F\left( \text{iω} \right)\ }{\text{d\ }\omega^{k}} = \ {( - i)}^{k}\mathcal{\ F\lbrack}t^{k}\text{\ f}\left( t \right)\rbrack$$

1. Przesunięcie argumentu funkcji

ℱ[f(t−t₀)] =   e^{− i ωt₀}  ℱ[f(t)] 

1. Przesunięcie argumentu transformaty

ℱ[e^{i ω₀ t} f(t)] = F(i(ω−ω₀))

1. ℱ−transformata pochodnej

Jeśli f^(k)(t) =  0   i   f^(k)(t) =  0   ,  k = 0, 1, …, n − 1 to   

ℱ[f⁽ⁿ⁾(t)] =  (iω)ⁿ ℱ[f(t)] 

Tabela

f(t)	ℱ[f(t)]
$$\left\{ \begin{matrix} 1,\ gdy\ \left| t \right| < a \\ 0,\ gdy\ \left| t \right| > a \\ \end{matrix} \right.\ $$	$$\frac{2\sin\text{a\ ω}}{\omega}$$
$$\left\{ \begin{matrix} 1,\ gdy\ 0 < t < 1 \\ 0\ ,\ w\ przeciwnym\ przypadku \\ \end{matrix} \right.\ $$	$$\frac{\sin\omega}{\omega} + \frac{i}{\omega}\ (\cos{\omega - 1)}$$
$$\left\{ \begin{matrix} 1 - \left| t \right|,\ \ gd\text{y\ }\left| t \right| < 1 \\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ gdy\ \left| t \right| > 1 \\ \end{matrix} \right.\ $$	$$\frac{2}{\omega^{2}}\ (1 - \cos{\omega)}$$
e^−c t   ,  t > 0; c > 0	$$\frac{1}{i\omega + c}$$
e^−c(t)	$$\frac{2c}{\omega^{2} + \ c^{2}}$$
e^−t2	$$\sqrt{\pi}\ e^{\frac{- \omega^{2}}{4}}$$
cos t, |t|<$\frac{\pi}{2}$	$$\frac{2\cos{\frac{\pi}{2}\omega}}{1 - \ \omega^{2}}$$
$$\frac{1}{1 + \ t^{2}}$$	π e^−|ω|
δ(t)	1

Splot funkcji

(f*g)(t) =  ∫_−∞^∞f(τ)g(t− τ)dτ

Własność splotu

ℱ[(f*g)(t)]= ℱ[f(t)] ℱ[g(t)]