# Laplace transform as rotation. In what space?

1. Mar 5, 2012

### mclaudt

Sorry for thumb terminology, I just would like to grasp the main idea, as a physicist, without unnecessary complications, associated with system of axioms and definitions.

Fourier transform can be seen as rotation of basis in space of all complex-valued functions from basis of delta-functions to a new basis of waves $e^{i\omega t}$.

Laplace transform can be seen as generalization of Fourier transform to complex frequencies. Is it correct to see the Laplace transform as a rotation of basis in space of complex-valued functions of complex argument, from delta-functions basis to a new basis of $e^{(\alpha + i\beta) t}$and, if it is so, what is the basis in that space, and why does summation go only along line $(-\infty, +\infty)$ in direct transform and $(\gamma - i\infty, \gamma + i\infty)$ in reverse transform?

2. Mar 5, 2012

### mclaudt

There is also an operation known as Wick rotation, it is a candidate to interpret the interconnection between Fourier and Laplace, it could be interesting if they both constitute some beautiful scheme.

3. Mar 5, 2012

### mclaudt

It seems that the key point is the analytical continuation. The raw form of Laplace transform is $F(s) = \int_{ℂ}e^{-su}f(u)du$, where $f(u)$ is analytical continuation of $f(t)$ to complex plane. And in that case the Laplace transform will be the rotation of basis in ${f(u)}$ space of complex functions of complex argument, just as Fourier transform is the rotation of basis in ${f(t)}$ space of complex functions of real argument, and integral is in both cases just a generalization of matrix multiplication to infinite dimension.

It's just guess, I'm not sure if it is correct to define $F(s)$ that way.

4. Mar 17, 2012