Why must X(s) occur in conjugate reciprocal pairs for a real x(t)?

[khdani]

Hello,
if i said that x(t) is real,
why X(s) must occur in conjugate reciprocal pairs ?

 

[CEL]

It does not.
If $$x(t) = e^{at}$$, then $$X(s) = \frac{1}{s-a}$$, which has a real pole at s = a.
In the other way, if x(t) is a sinusoid, then X(s) will have complex conjugate poles.

 

[Mene]

The poles of the Laplace transform represent the values of s for which the transform is undefined or infinite. In the case of a real x(t), the Laplace transform X(s) must occur in conjugate reciprocal pairs because of the way the Laplace transform is defined. The transform is an integral over the complex plane, and the integration path must be chosen carefully to avoid crossing any poles. For a real x(t), the integration path is typically chosen to be along the real axis. However, if there is a pole on the real axis, the integral is undefined. Therefore, in order for the integral to be well-defined, the poles must occur in conjugate reciprocal pairs, with one pole on the upper half of the complex plane and its conjugate on the lower half. This ensures that the integration path avoids all poles and the integral is well-defined.