This is a question I encountered in my own research on Generalized Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the basis measure for this family is $ $ L\left( \theta\right) =\left( \cos \theta\right) ^{-\sigma}$ $ where $ \sigma >0$ is the scale parameter and $ \vert \theta \vert < \pi/2$Read more