Common-formulas

常用拉普拉斯变换

\begin{matrix} (1) & \begin{array}{ccc} 序号 & 拉氏变换 F (s) & 时间函数 f (t) \\ 1 & 1 & δ (t) \\ 2 & \frac{1}{1 - e^{- 5}} & δ_{T} (t) = \sum_{n = 0}^{\infty} δ (t - n T) \\ 3 & \frac{1}{s} & 1 (t) \\ 4 & \frac{1}{s^{2}} & t \\ 5 & \frac{1}{s^{3}} & \frac{t^{2}}{2} \\ 6 & \frac{1}{s^{n + 1}} & \frac{t^{n}}{n!} \\ 7 & \frac{1}{s + a} & e^{- a t} \\ 8 & \frac{1}{(s + a)^{2}} & t e^{- a t} \\ 9 & \frac{a}{s (s + a)} & 1 - e^{- a t} \\ 10 & \frac{b - a}{(s + a) (s + b)} & e^{- a t} - e^{- b t} \\ 11 & \frac{ω}{s^{2} + ω^{2}} & \sin ω t \\ 12 & \frac{s}{s^{2} + ω^{2}} & \cos ω t \\ 13 & \frac{ω}{(s + a)^{2} + ω^{2}} & e^{- a t} \sin ω t \\ 14 & \frac{s + a}{(s + a)^{2} + ω^{2}} & e^{- a t} \cos ω t \\ 15 & \frac{1}{s - (1 / T) \ln a} & a^{t / T} \end{array} \end{matrix}

$L[a f(t)]=a F(s)$

$L\left[f_{1}(t) \pm f_{2}(t)\right]=F_{1}(s) \pm F_{2}(s)$

微分定理一般形式:

\begin{matrix} (2) & \begin{matrix} L [\frac{d f (t)}{d t}] = s F (s) - f (0) \\ L [\frac{d^{2} f (t)}{d t^{2}}] = s^{2} F (s) - s f (0) - f^{'} (0) \\ ⋮ \\ L [\frac{d^{n} f (t)}{d t^{n}}] = s^{n} F (s) - \sum_{k = 1}^{n} s^{n - k} f^{(k - 1)} (0) \\ f^{(k - 1)} (t) = \frac{d^{k - 1} f (t)}{d t^{k - 1}} \end{matrix} \end{matrix}

微分定理（当初始条件为0的时候）:

\begin{matrix} (3) & \begin{matrix} L [\frac{d^{n} f (t)}{d t^{n}}] = s^{n} F (s) \end{matrix} \end{matrix}

积分定理一般形式:

\begin{matrix} (4) & \begin{matrix} L [\int f (t) d t] = \frac{F (s)}{s} + \frac{{[\int f (t) d t]}_{t - 0}}{s} \\ L [\iint f (t) (d t)^{2}] = \frac{F (s)}{s^{2}} + \frac{{[\int f (t) d t]}_{t_{-} 0}}{s^{2}} + \frac{{[\iint f (t) (d t)^{2}]}_{t_{-} 0}}{s} \\ L [\int \dots \int f (t) (d t)^{n}] = \frac{F (s)}{s^{n}} + \sum_{k = 1}^{n} \frac{1}{s^{n - k + 1}} {[\int \dots \int f (t) (d t)^{n}]}_{t - 0} \end{matrix} \end{matrix}

积分定理（当初始条件为0的时候）:

\begin{matrix} (5) & L [\int_{\dots} \int f (t) (d t)^{n}] = \frac{F (s)}{s^{n}} $ $ \end{matrix}

\begin{matrix} (6) & \begin{matrix} U_{L} = L \frac{d i}{d t} \\ C \frac{d u_{C}}{d t} = i \end{matrix} \end{matrix}