Laplace Transform

Laplace Transform - Definition

Laplace Transform Pair:

1. The Laplace Transform 𝐿{·} of a function 𝑓(𝑡) is defined as:
   • $\overline{f}(s)=L\{f(t)\}(s)={\int }_0^{\infty }{e}^{-st}f(t)dt$

   • derivation

     As defined in the excerpt above, let:

     • $F(t)=f(t)e^{-𝜁t}H(t)$

     The Laplace transform of 𝑓(𝑡) is the Fourier transform of 𝐹(𝑡):

     • $\overline{f}(s)=L\{f(t)\}(s)=F\{F(t)\}(s)=\hat{F}(s)$

     So let’s compute the Fourier transform of 𝐹.

     Let 𝐹ˆ(𝑤) be the Fourier transform of 𝐹(𝑡):

     • $\hat{F}(w)={\int }_{-\infty }^{\infty }F(t)e^{-iwt}dt\text{by definition of Fourier Transform}$
     • $\hat{F}(w)={\int }_{-\infty }^{\infty }f(t)e^{-𝜁t}H(t)e^{-iwt}dt\text{by substitution of }F(t)\text{ as defined in excerpt}$
     • $\hat{F}(w)={\int }_0^{\infty }f(t)e^{-𝜁t}{e}^{-iwt}dt\text{by definition of heavyside function }H(t)$
     • $\hat{F}(w)={\int }_0^{\infty }f(t)e^{-(𝜁+iw)t}dt$
     • $\hat{F}(s)={\int }_0^{\infty }f(t)e^{-st}dt\text{by variable substitution }s=(𝜁+iw)$

     Since the Laplace transform of 𝑓(𝑡) is the Fourier transform of 𝐹(𝑡), then:

     • $\overline{f}(s)=L\{f(t)\}(s)=F\{F(t)\}(s)=\hat{F}(s)=\hat{F}(s)={\int }_0^{\infty }f(t)e^{-st}dt$

2. The Inverse Laplace Transform 𝐿^-1{·} of a function 𝑓ˆ(𝑡) is defined as:
   • $f(t)=L^{-1}\{\overline{f}(s)\}(t)=\frac{1}{2\pi i}{\int }_{𝜁-i\infty }^{𝜁+i\infty }{e}^{st}\overline{f}(s)ds$

   • derivation

     As defined in the excerpt above, let:

     • $F(t)=f(t)e^{-𝜁t}H(t)$

     So let’s compute the Inverse Fourier transform of 𝐹ˆ.

     Let 𝐹(𝑡) be the Inverse Fourier transform of 𝐹ˆ(𝑤):

     • $F(t)=F^{-1}\{\hat{F}(w)\}(t)$
     • $F(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\hat{F}(w)e^{iwt}dw\text{by definition of Inverse Fourier Transform }{F}^{-1}$
     • $F(t)e^{𝜁t}=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\hat{F}(w)e^{iwt}{e}^{𝜁t}dw$
     • $f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\hat{F}(w)e^{iwt}{e}^{𝜁t}dw\text{ because }F(t)=f(t)e^{-𝜁t}H(t)$
     • $f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\hat{F}(w)e^{(𝜁+iw)t}dw$
     • Let’s define:
       • $s=𝜁+iw$
       • $ds=idw⟹dw=\frac{1}{i}ds$
       • The Laplace transform of 𝑓(𝑡) is the Fourier transform of 𝐹(𝑡):
       • $\overline{f}(s)=L\{f(t)\}(s)=F\{F(t)\}(w)=\hat{F}(w)$
     • $f(t)=\frac{1}{2\pi }{\int }_{𝜁-i\infty }^{𝜁+i\infty }\overline{f}(s)e^{st}\frac{1}{i}ds\text{ via substitution }$
     • $f(t)=\frac{1}{2\pi i}{\int }_{𝜁-i\infty }^{𝜁+i\infty }\overline{f}(s)e^{st}ds$

Laplace Transform - Examples

See: Laplace Transform - Examples

Laplace Transform - Properties

Laplace transforms are linear operators:

• $L\{a\ast f(t)+b\ast g(t)\}=a\ast L\{f(t)\}+b\ast L\{g(t)\}$

Laplace Transform - Use Cases

• used in solving differential equations:
  • transforms PDE into ODE
  • transforms ODE into algebraic expressions
• used in Control Theory

Resources

• https://www.khanacademy.org/math/differential-equations/laplace-transform
• The Laplace Transform: A Generalized Fourier Transform