Skip to main content

Mathematical Methods of Physics: For Undergraduate

Sunil Karna

Subsection 6.3.1 Properties of Laplace Transforms

The Laplace transform has several properties that make it a powerful tool for solving differential equations and studying linear time-invariant systems. Some of the key properties include:
  1. Linearity: \(\mathscr{L}\{af(t) + bg(t)\} = aF(s) + bG(s)\text{,}\) where ’a’ and ’b’ are constants.
  2. Shifting: \(\mathscr{L}\{f(t - a)\} = e^(-as) * F(s)\text{,}\) which represents a time shift of ’a’ units.
  3. Differentiation: \(\mathscr{L}\{f'(t)\} = sF(s) - f(0)\text{,}\) where f’(t) represents the derivative of f(t).
  4. Integration: \(\mathscr{L}\{ \int^{t}_{0} f(\tau) \,d\tau \} = 1/s * F(s)\text{,}\) where the integral represents the indefinite integral of f(t) with respect to t.
  5. Convolution: \(\mathscr{L}{f(t) * g(t)} = F(s) * G(s)\text{,}\) where * denotes convolution.

Subsubsection 6.3.1.1 Linear Property

\begin{equation*} \mathscr{L}\{af_{1}(t)+bf_{2}(t)\} =a \mathscr{L}\{f_{1}(t)\} + b \mathscr{L}\{f_{2}(t)\} \end{equation*}

Proof.

\begin{equation*} \mathscr{L}\{af_{1}(t)+bf_{2}(t)\} = \int\limits_{0}^{\infty}e^{-st}\left[af_{1}(t)+bf_{2}(t)\right] \end{equation*}
\begin{equation*} =a\int\limits_{0}^{\infty}e^{-st}f_{1}(t)\,dt+b\int\limits_{0}^{\infty}e^{-st}f_{2}(t)\,dt \end{equation*}
\begin{equation*} =a \mathscr{L}\{f_{1}(t)\} + b \mathscr{L}\{f_{2}(t)\} \end{equation*}

Subsubsection 6.3.1.2 First-Shifting Property

If \(\mathscr{L}\{af(t)\}=F(s),\) then \(\mathscr{L}\{e^{at}f(t)\} = F(s-a)\text{.}\)

Proof.

\begin{equation*} \mathscr{L}\{e^{at}f(t)\} = \int\limits_{0}^{\infty}e^{-st}e^{at}f(t)\,dt \end{equation*}
\begin{equation*} = \int\limits_{0}^{\infty}e^{-(s-a)t}f(t)\,dt =\int\limits_{0}^{\infty}e^{-rt}f(t)\,dt =F(r) = F(s-a) \end{equation*}
[\(\because s-a =r \)]

Subsubsection 6.3.1.3 Inverse Laplace Transforms

If \(\mathscr{L}[f(t)] =F(s)\) then \(f(t)\) is called the inverse Laplace Transform of \(F(s)\) and we write \(f(t) = \mathscr{L}^{-1}F(s)\text{.}\) Here \(\mathscr{L}^{-1}\) is called the inverse Laplace Transform operator.
PrevTopNext