Laplace Transform: Heaviside Function Classroom

98.1 Heaviside Unit Step Function

The Heaviside unit step function, denoted as $H(t-c)$ or $u(t-c)$, is a discontinuous function whose value is zero for arguments less than zero and one for arguments greater than or equal to zero.

It is defined as:

where $c$ is the point at which the step occurs. This function is fundamental in engineering for representing signals that switch on or off at a specific time, such as a voltage applied to a circuit at $t=0$ or a force applied at $t=c$.

98.2 Transform of $H(t-c)$

The Laplace Transform of a simple shifted Heaviside unit step function $H(t-c)$ is given by:

This formula is derived directly from the definition of the Laplace Transform by considering the limits of integration from $c$ to $\infty$ instead of $0$ to $\infty$, since $H(t-c)$ is zero for $t < c$.

98.3 Transform of $H(t-c)f(t-c)$ (Second Shifting Theorem)

The Second Shifting Theorem (or Time Shifting Property) is crucial for finding the Laplace Transform of functions that are "switched on" at a specific time $c$. It states that if $\mathcal{L}\{f(t)\} = F(s)$, then:

This theorem indicates that a time delay of $c$ in the time domain corresponds to multiplication by $e^{-cs}$ in the s-domain.

98.4 Inverse Transforms (Heaviside)

To find the inverse Laplace Transform of functions involving $e^{-cs}F(s)$, we reverse the Second Shifting Theorem:

where $f(t) = \mathcal{L}^{-1}\{F(s)\}$. This means that if you have an exponential term $e^{-cs}$ multiplying a function $F(s)$ in the s-domain, the inverse transform will be the original time-domain function $f(t)$ shifted by $c$ and multiplied by the Heaviside step function $H(t-c)$.