# Formulae for the Extended Laplace Integral and their Statistical Applications

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 44, (2014), **Pages** 61-74

**Thanh T. Tran**

Department of Statistics,

University of Auckland

**Thomas W. Yee**

Department of Statistics,

University of Auckland

**Garry J. Tee**

Department of Mathematics,

University of Auckland

mailto:tee@math.auckland.ac.nz

**Abstract** We propose an extension of the Laplace integral and derive formulae to evaluate it over finite intervals. This integral is a generalization of the gamma function and modified Bessel function of the third kind. Consequently, our results provide not only formulae in terms of the complementary error function to evaluate the incomplete gamma functions, but also those for the lower and upper incomplete Bessel functions. Statistically, our formulae allow for the derivation of the distribution functions of the generalized inverse Gaussian (GIG) and gamma distributions in terms of the complementary error function, which have not been documented in the literature.

**Keywords** Upper and lower incomplete Bessel functions, incomplete gamma functions, complementary error function.

**Classification** (MSC2000) Primary 33C10, 33C15, 33E20, 33F05, 65D20. Secondary 33B20.