14 Laplace
Laplace Transformation¶
Converting differential calculus into algebra
flowchart LR
DE -->
|LT| ae[Algebraic] -->
|Solving| sa[Algebraic Solution] -->
|ILT| sd[DE Solution] -.->
DELT Function¶
Laplace Transform
\[ \begin{aligned} L\{ f(t) \} &= \int\limits_0^\infty e^{-st} f(t) \cdot dt \quad \text{ (or a function of } x)\\ &= F(s) \end{aligned} \]
| \(f(t)\) | Time Domain Function |
| \(s\) | Laplace Variable (real/complex) |
| \(F(s)\) | Laplace Domain Function |
ILT Function¶
Inverse Laplace Transform
\[ L^{-1} \{ F(s) \} = f(t) \]
Basic Rules¶
| Situation | LT | ILT |
|---|---|---|
| Constant Coeffient | \(L\Big(k f(t) \Big) = k L(t)\) | \(L^{-1}(k s) = k L^{-1} \Big( F(s) \Big)\) |
| Sum | \(L \Big( f(t) \pm g(t) \Big) = L \Big( f(t) \Big) \pm L \Big( g(t) \Big)\) | \(L^{-1} \Big( F(s) \pm G(s) \Big) = L^{-1} \Big( F(s) \Big) \pm L^{-1} \Big( G(s) \Big)\) |
LT of Standard Functions¶
| \(f(t)\) | \(L\Big( f(t) \Big)\) |
|---|---|
| \(1\) | \(\frac{1}{s}\) |
| \(k\) | \(\frac{k}{s}\) |
| \(e^{at}\) | \(\frac{1}{s\textcolor{orange}{-}a}\) |
| \(\cos(at)\) | \(\frac{s}{s^2 + a^2}\) |
| \(\sin(at)\) | \(\frac{a}{s^2 + a^2}\) |
| \(\cosh(at)\) | \(\frac{s}{s^2 - a^2}\) |
| \(\sinh(at)\) | \(\frac{a}{s^2 - a^2}\) |
| \(t^n\) | $\begin{cases} \dfrac{n!}{s^{n+1}}, & n \le 0 \ |
| \dfrac{\Gamma(n+1)}{s^{n+1}}, & \text{otherwise} \end{cases}$ where \(\Gamma\) is gamma function | |
| \(e^{at} f(t)\) (exponent shifting rule) | \(F(s \textcolor{orange}{-} a) = \Big\{ F(s) \Big\}_{s \to s-a}\) |
| \(u_a(t)\) | \(\frac{e^{-as}}{s}\) |
| \(\delta (t)\) | \(1\) |
Unit Step Function¶
\[ u_a (t) = \begin{cases} 0, & t < a \\ 1, & t \ge a \end{cases} \]
Unit Impulse Function¶
\[ \delta (t) = \lim_{\epsilon \to 0} f_\epsilon(t) \]
\[ f_\epsilon(t) = \begin{cases} \dfrac{1}{\epsilon}, & 0 \le t \le \epsilon \\ 0, & t > \epsilon \end{cases} \]
\[ \begin{aligned} L\Big( \delta(t) \Big) &= \lim_{\epsilon \to 0} L\Big( f_\epsilon (t) \Big) \\ &= \lim_{\epsilon \to 0} \left[ \int\limits_0^\infty e^{-st} f_\epsilon(t) \cdot dt \\ \right] \\ & \dots \\ &= 1 \end{aligned} \]
Sum of GP¶
\[ \sum GP = \frac{a}{1-r} \]
Gamma Function¶
\[ \Gamma(x) = \int_0^\infty e^{-x} x^{n-1} dx \]
Properties¶
\[ \begin{aligned} \Gamma \left(\frac{1}{2} \right) &= \sqrt{\pi} \\ \Gamma(n) &= (n-1)! \\ &= (n-1) \cdot \Gamma(n-1) \\ n! &= \Gamma (n+1) \\ \end{aligned} \]
IDK¶
When doing nested transformations, do it as Part 1 and \(f(part 1)\) like how you did it for grade 12 integrals \(I_1 + I_2\)