Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by

$$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$

I visualize the complex unit factor $e^{-2 \pi i \, \omega \,x}$ as a rotating probe that sweeps the unit circle with frequency $\omega$, as $x$ sweeps from $-\infty$ to $\infty$, and thereby "picks out" those features of the curve $f(x) \subset {\mathbb C}$ that contribute to the component of its spectrum at frequency $\omega$. (The integral sums all these contributions to produce the $\omega$ component of $f\;$'s spectrum.)

Is it possible to give a similar "intuitive interpretation" of what the Laplace transform $$\mathcal{L}(f)(s) = \int_{0}^\infty e^{-sx}f(x)dx, \;\;\; s\in {\mathbb C}$$ is doing?

share|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.