Abstract
A parametric family of complex valued functions in the form of quadratic exponents are defined starting from the real valued Gaussian function. The family of functions include the quadratic phase function, which is also called the two-sided complex chirp. It is proven using contour integrals on the complex plane that the Laplace transforms of these functions are also complex valued quadratic exponents; the region of convergence of the Laplace transform include the entire <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> -plane for finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$|s|$ </tex-math></inline-formula> for a range of values of the defining parameter of the family. Fourier transforms are also presented as the special cases of the Laplace transform.
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