In mathematics, the Laplace transform is used to convert a function of time into a function of the complex frequency s. In LaTeX, you can represent the Laplace transform using the \mathcal{L} command.

Laplace Transform in LaTeX

In mathematics, the Laplace transform is used to convert a function of time into a function of the complex frequency s. In LaTeX, you can represent the Laplace transform using the \mathcal{L} command.

Using the \mathcal{L} command

To write the Laplace transform of a function $f(t)$ in LaTeX, use the \mathcal{L} command combined with the function in curly braces and the transform variable (s) in parentheses. For example:

$$
\mathcal{L}\{f(t)\}(s) = \int_0^{\infty} e^{-st} f(t) dt
$$
\[\mathcal{L}\{f(t)\}(s) = \int_0^{\infty} e^{-st} f(t) dt\]

This represents the Laplace transform of the function $f(t)$.

Examples

Here are some examples of using the \mathcal{L} command to represent Laplace transforms in LaTeX:

  1. Laplace transform of an exponential function:
$$
\mathcal{L}\{e^{at}\}(s) = \frac{1}{s-a}
$$
\[\mathcal{L}\{e^{at}\}(s) = \frac{1}{s-a}\]

This represents the Laplace transform of the exponential function $e^{at}$.

  1. Laplace transform of a periodic function:
$$
\mathcal{L}\{\cos(\omega t)\}(s) = \frac{s}{s^2 + \omega^2}
$$
\[\mathcal{L}\{\cos(\omega t)\}(s) = \frac{s}{s^2 + \omega^2}\]

This represents the Laplace transform of the periodic cosine function $\cos(\omega t)$.

These examples show how to use the \mathcal{L} command to represent Laplace transforms in LaTeX.