Some Laplace transforms and integral representations for parabolic cylinder functions and error functions

Year 2021, Volume: 50 Issue: 1, 63 - 78, 04.02.2021

Dirk Veestraeten

https://doi.org/10.15672/hujms.612642

Abstract

This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently specialized for products of the error function and its complement thereby yielding new integral representations for products of the latter two functions. The transforms that are derived in this paper also allow to correct two inverse Laplace transforms that are widely reported in the literature and subsequently uses one of the corrected expressions to obtain two new definite integrals for the generalized hypergeometric function.

Keywords

confluent hypergeometric function, convolution theorem, error function, Gaussian hypergeometric function, generalized hypergeometric function, Laplace transform, parabolic cylinder function

References

• [1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1972.
• [2] A. Ali, T. Lebel and A. Mani, Rainfall Estimation in the Sahel. Part I: Error Function, J Appl. Meteorol. 44, 1691–1706, 2005.
• [3] J.V. Beck, K.D. Cole, A. Haji-Sheikh and B. Litkouhi, Heat Conduction Using Green’s Functions, Hemisphere Publishing Corporation, London, 1992.
• [4] J.-F. Bercher and C. Vignat, On minimum Fisher information distributions with restricted support and fixed variance, Inform. Sci. 179, 3832–3842, 2009.
• [5] R.M. Capocelli and L.M. Ricciardi, Diffusion Approximation and First Passage Time Problem for a Model Neuron, Kybernetik 8 (6), 214–223, 1971.
• [6] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford, 1959.
• [7] R. Combescot and T. Dombre, Superfluid current in $^{3}$He-$A$ at $T=0$, Phys. Rev. B 28 (9), 5140–5149, 1983.
• [8] J.L. deLyra, S.K. Foong and T.E. Gallivan, Finite lattice systems with true critical behavior, Phys. Rev. D 46 (4), 1643–1657, 1992.
• [9] L. Durand, Nicholson-type integrals for products of Gegenbauer functions and related topics, in: Theory and Applications of Special Functions, 353–374, Academic Press, New York, 1975.
• [10] Á. Elbert and M.E. Muldoon, Inequalities and monotonicity properties for zeros of Hermite functions, Proc. Roy. Soc. Edinburgh Sect. A 129, 57–75, 1999.
• [11] Á Elbert and M.E. Muldoon, Approximations for zeros of Hermite functions, in: Special Functions and Orthogonal Polynomials, Contemporary Mathematics 471, 117–126, American Mathematical Society, Providence, 2008.
• [12] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.
• [13] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.
• [14] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1954.
• [15] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., John Wiley & Sons, New York, 1968.
• [16] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 8th ed., edited by D. Zwillinger and V. Moll, Academic Press, New York, 2014.
• [17] Yu.P. Kalmykov, W.T. Coffey and J.T. Waldron, Exact analytic solution for the correlation time of a Brownian particle in a doublewell potential from the Langevin equation, J. Chem. Phys. 105 (5), 2112–2118, 1996.
• [18] C. Malyshev, Higher corrections to the mass current in weakly inhomogeneous superfluid $^{3}$He-$A$, Phys. Rev. B 59 (10), 7064–7075, 1999.
• [19] E.W. Ng and M. Geller, A Table of Integrals of the Error Functions, J. Res. NBS 73B (1), 1–20, 1969.
• [20] Y. Nie and V. Linetsky, Sticky reflecting Ornstein–Uhlenbeck diffusions and the Vasicek interest rate model with the sticky zero lower bound, Stoch. Models, forthcoming.
• [21] F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer–Verlag, Berlin, 1973.
• [22] K. Oldham, J. Myland and J. Spanier, An Atlas of Functions, 2nd ed., Springer– Verlag, Berlin, 2009.
• [23] J.K. Patel and C.B. Read, Handbook of the Normal Distribution, Marcel Dekker, New York, 1982.
• [24] A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. More Special Functions, Vol. 3, Gordon and Breach, New York, 1990.
• [25] A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. Direct Laplace Transforms, Vol. 4, Gordon and Breach, New York, 1992.
• [26] A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. Inverse Laplace Transforms, Vol. 5, Gordon and Breach, New York, 1992.
• [27] D. Veestraeten, Some integral representations and limits for (products of) the parabolic cylinder function, Integr. Transf. Spec. F. 27 (1), 64–77, 2016.
• [28] D. Veestraeten, An integral representation for the product of parabolic cylinder functions, Integr. Transf. Spec. F. 28 (1), 15–21, 2017.
• [29] E.T. Whittaker, On the Functions Associated with the Parabolic Cylinder in Harmonic Analysis, Proc. Lond. Math. Soc. 35, 417–427, 1902.
• [30] T.V. Zaqarashvili and K. Murawski, Torsional oscillations of longitudinally inhomogeneous coronal loops, Astron. Astrophys. 470, 353–357, 2007.