Lecture

Definition of Laplace Transform

• $\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st}f(t) \, dt$

  L{f(t)}=∫0∞​e−stf(t)dt

• Domain and convergence criteria

• Piecewise continuity and exponential order conditions

Common Laplace Transforms

• $\mathcal{L}\{1\} = \frac{1}{s}$

  L{1}=s1​

• $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$

  L{tn}=sn+1n!​

• $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$

  L{eat}=s−a1​

• And others

Properties of Laplace Transform:

• Linearity:

  $\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$

  L{af(t)+bg(t)}=aF(s)+bG(s)

• First Shifting Theorem (Time Shift):

  $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$

  L{eatf(t)}=F(s−a)

• Differentiation in Time Domain:

  $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$

  L{f′(t)}=sF(s)−f(0)

• Integration in Time Domain

• Convolution Theorem

  • Initial and Final Value Theorems https://studio.youtube.com/video/YW0PlNLX1j8/edit