Inverse Laplace Transforms Classroom

97.1 Definition of Inverse Laplace Transform

The Inverse Laplace Transform, denoted as $\mathcal{L}^{-1}\{F(s)\}$, is the operation that converts a function in the complex frequency domain ($s$-domain) back to a function in the time domain ($t$-domain).

If $\mathcal{L}\{f(t)\} = F(s)$, then $\mathcal{L}^{-1}\{F(s)\} = f(t)$.

While the forward Laplace Transform is defined by an integral, the inverse transform is generally more complex and often involves techniques like partial fraction decomposition, convolution theorem, or looking up tables of common transforms.

97.2 Simple Function Inverses

For many common functions, finding the inverse Laplace Transform involves recognizing the form of $F(s)$ and matching it to entries in a table of Laplace Transforms. This is essentially reversing the process of finding the forward transform.

• $\mathcal{L}^{-1}\left\{\frac{1}{s}\right\} = 1$
• $\mathcal{L}^{-1}\left\{\frac{n!}{s^{n+1}}\right\} = t^n$
• $\mathcal{L}^{-1}\left\{\frac{1}{s-a}\right\} = e^{at}$
• $\mathcal{L}^{-1}\left\{\frac{a}{s^2 + a^2}\right\} = \sin(at)$
• $\mathcal{L}^{-1}\left\{\frac{s}{s^2 + a^2}\right\} = \cos(at)$

The linearity property also applies to inverse Laplace Transforms: $\mathcal{L}^{-1}\{aF(s) + bG(s)\} = a\mathcal{L}^{-1}\{F(s)\} + b\mathcal{L}^{-1}\{G(s)\}$.

97.3 Inverse Transforms (Partial Fractions)

For rational functions $F(s)$ that are more complex than simple elementary forms, partial fraction decomposition is a crucial technique. It allows us to break down a complicated rational function into a sum of simpler fractions, each of which corresponds to a known inverse Laplace Transform.

Steps:

1. Factor the denominator of $F(s)$.
2. Set up the partial fraction expansion based on the factors (linear, repeated linear, irreducible quadratic).
3. Solve for the unknown coefficients in the partial fractions.
4. Take the inverse Laplace Transform of each simpler fraction.

97.4 Poles & Zeros of Laplace Transforms

In the context of Laplace Transforms, poles and zeros are critical for understanding the behavior of systems in the frequency domain and their corresponding time-domain responses.

• Poles: The values of $s$ for which the Laplace Transform $F(s)$ becomes infinite (i.e., the roots of the denominator of $F(s)$). Poles dictate the form of the time-domain response $f(t)$ (e.g., exponential decay, oscillations).
• Zeros: The values of $s$ for which the Laplace Transform $F(s)$ becomes zero (i.e., the roots of the numerator of $F(s)$). Zeros influence the magnitude and phase of the frequency response but do not directly determine the stability or general form of the time response in the same way poles do.

The location of poles in the complex $s$-plane is particularly important for stability analysis in control systems and circuit theory.