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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.
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.
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 $f(t)$.
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 $e^{at}$.
2. 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.