Some closed form expressions for the generalized secant integrals

Abstract

The generalized secant integrals of the form I a ( ψ , b ) = b a ∫ 0 ψ exp ( - b sec θ ) ( sec θ ) a d θ for b > 0 and 0 < ψ ⩽ π / 2 arise frequently in shielding problems. The recent papers by Michieli [1998. Point kernel calculations of dose fields from line sources using expanded polynomial form of buildup factor data: generalized secant integral series representations. Radiat. Phys. Chem. 51, 121–128; 2001. Some properties of generalized secant integrals: extended definition and recurrence relations. Radiat. Phys. Chem. 60, 551–554] and Guseinov and Mamedov [2004. Calculation of generalized secant integral using binomial coefficients. Appl. Radiat. Isot. 60, 689–692] provided certain infinite series representations as well as recurrence relations for computing these integrals. However, no closed form expressions for I a ( ψ , b ) have been known except when both a and ψ are fixed. In this note, we provide several closed form expressions for I a ( ψ , b ) applicable for a wide range of values of a and b. We establish their numerical accuracy over the methods presented in Michieli [1998. Point kernel calculations of dose fields from line sources using expanded polynomial form of buildup factor data: generalized secant integral series representations. Radiat. Phys. Chem. 51, 121–128; 2001. Some properties of generalized secant integrals: extended definition and recurrence relations. Radiat. Phys. Chem. 60, 551–554]. Finally, a simple computer program is provided for I a ( ψ , b ) that could be used widely.