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.

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:

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

$$\mathcal{L}\{f(t)\}(s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(t)dt$$

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.