# What's the connection between the Laplace transform and the Fourier transform?

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition.

Are there deep connections between these two transforms? The formulaic connection is clear, but is there something deeper?

(Maybe the answer will involve spectral theory?)

• In what sense does the Laplace transform give a power-series decomposition? I don't understand the relationship between this question and the question you linked to. Commented Mar 18, 2011 at 23:03
• The obvious link is more natural and pertinent, I think, that the question you linked. en.wikipedia.org/wiki/Laplace_transform#Fourier_transform Commented Mar 19, 2011 at 0:00
• @Qiaochu Yuan A power series says what constants $\vec{a}$ will make $\sum a_i x^i = f(x)$. The Laplace (Mellin) transform says what function $a(i)$ will make $\int a(i) x^i = f(x)$. In the linked Q, @Christian Blatter's answer gives $F(phi) = \sum_{k=0}^n a_k e^{i k \phi}$. Commented Mar 19, 2011 at 0:05
• Laplace can only mutiply or divide the signals. Fourier can only add or subtract the signals
– user88296
Commented Jul 30, 2013 at 14:58
• I'd love to see a more precise version of the answer. Some people are flagging it as "not an answer," but it seems like an incomplete and potentially interesting answer. Commented Jul 30, 2013 at 15:43