A Note on Error Functions and Some Related Special Functions

Year 2026, Volume: 10 Issue: Special, 9 - 14, 05.03.2026

Hüseyin Irmak

https://izlik.org/JA57DH64FC

Abstract

In this note, priority is given to presenting essential information on the fundamental error functions, which play particularly different roles in the natural and engineering sciences, as well as some special functions closely related to them. Subsequently, various special relationships among these functions are highlighted, and finally, some selected applications of these relationships — emphasized through several references — are also proposed for interested readers.

Keywords

Error functions , Faddeeva function , Gaussian function and intgeral , Series expansion , Dawson integral

References

• [1] A. Erdelyi, W. Magnus, F. Oberhettinge, and F. G. Tricomi, Higher Transcendental Functions, I. McGraw-Hill, New York, Toronto, London, 1953.
• [2] E. D. Rainville, Special Functions, Macmillan: New York, NY, USA, 1960.
• [3] N. N. Lebedev, Special Functions and Their Applications, translated by R. A. Silverman, New Jersey, Prentice-Hall Inc, 1965.
• [4] W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics, New York: Springer -Verlag, 1966.
• [5] M. Abramowitz and I. A. Milton, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, New York, Dover, 1972.
• [6] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1999.
• [7] G. A. Korn and T. M. Korn, Mathematical handbook for scientists and engineers: Definitions, theorems and formulas for reference and review, Dover Publications, 2000.
• [8] R. Beals and R. Wong, Special Functions: A Graduate Text, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2010.
• [9] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
• [10] P. Henrici, Applied and Computational Complex Analysis, Wiley, New York, 1977.
• [11] R. E. Greene and S. G. Krantz, Function theory of one complex variable, New York, Wiley, 1997.
• [12] J. B. Conway, Functions of One Complex Variable. II, Graduate Texts in Mathematics, Springer-Verlag, New York, 1995
• [13] M. Stein and R. Shakarchi, Lecture in Analysis II: Complex Analysis, Princeton University Press, Princeton, USA, 2003.
• [14] B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, London, 1977.
• [15] A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics: A Unified Introduction with Applications, Birkhäuser Boston, MA, 1988.
• [16] B. Davies, Integral Transforms and Their Applications, Springer, 2001.
• [17] B. Davies, Integral transforms and their applications, Springer, New York, 2002.
• [18] D. Lokensth and B. Dambaru, Integral Transformations and Their Applications, Boca Raton, CRC Press, 2007.
• [19] K. Knopp, Theory and Application of Infinite Series, London, Blackie & Son, 1954.
• [20] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2007.
• [21] C. Chiarella and A. Reiche, "On the evaluation of integrals related to the error function", Mathematics of Computation, vol. 22, pp. 137-143, 1968.
• [22] W. Ng and M. Geller, A table of integrals of the error functions," Journal of Research of the National Bureau of Standards, vol. 73 B, pp. 1-20, 1969.
• [23] W. Gautschi, "Efficient computation of the complex error function," SIAM Journal on Numerical Analysis, vol. 7, pp. 187-198, 1970.
• [24] H. Alzer, "Error functions inequalities", Adv. Comput. Math., vol. 33, pp. 349-379, 2010.
• [25] F. Schreier, "The Voigt and complex error function: A comparison of computational methods", Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 48, no. 5-6, pp. 743-762, 1992.
• [26] S. J. McKenna, "A method of computing the complex probability function and other related functions over the whole complex plane", Astrophysics and Space Science, vol. 107, pp. 71-83, 1984.
• [27] J.A.C. Weideman, "Computation of the Complex Error Function", SIAM J. Numerical Analysis, vol. 48, no. 5, 1994, pp. 1497-1518. Online: http://www.jstor.org/stable/2158232
• [28] N. J. Fine, Basic Hypergeometric Series and Applications, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1988.
• [29] L. J. Slater, Generalized hypergeometric functions, Cambridge, UK: Cambridge, 1966.
• [30] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. New York, 2020.
• [31] W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, Stechert – Hafner, New York, 1964.
• [32] Y. L. Luke, The Special Functions and their Approximations, vols. 1 and 2. New York: Academic Press, 1969.
• [33] W. Feller, An Introduction to Probability Theory and Its Applications, New York, John Wiley, 1971.
• [34] L. C. Andrews, Special Functions for Engineers and Applied Mathematics, New York, MacMillan Comp., 1984.
• [35] J. Spanier and K. B. Oldham, Dawson's Integral. Ch. 42 in An atlas of Functions, Washington, DC: Hemisphere, 1987.
• [36] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists, Springer, New York, NY, USA, 2008.
• [37] S. Sykora, Dawson’s integral approxima-tions, 2012, Recuperado en agosto de 2019 de:http://www.ebyte.it/library/codesnippets/DawsonIntegralApproximations.html. doi: https://doi.org/10.3247/SL4soft12.001.
• [38] J. P. Boyd, "Evaluating of Dawson's integral by solving its differential equation using orthogonal rational Chebyshev functions", Applied Mathematics and Computation, vol. 204, no. 2, pp. 914-919, 2008.
• [39] W. J. Cody, K. A. Paciorek and H. C. Thacher, "Chebyshev approximations for Dawson's integral", Mathematics of Computation, vol 24, pp. 171-178, 1970.
• [40] F. G. Lether, "Constrained near-minimax rational approximations to Dawson's integral", Applied Mathematics and Computation, vol. 88, no. 2, pp. 267-274, 1997.
• [41] F. G. Lether, "Shifted rectangular quadrature rule approximations to Dawson's integral F(x)", Journal of Computational and Applied Mathematics, vol. 92, no. 2, pp. 97-102, 1998.
• [42] D. Franta, D. Nečas, A. Giglia, P. Franta and I. Ohlídal, "Universal dispersion model for characterization of optical thin films over wide spectral range: Application to magnesium fluoride", Applied Surface Science, vol. 421, no. 1, pp. 424-429, 2017. doi: https://doi.org/10.1016/j.apsusc.2016.09.149
• [43] S.M Abrarov and B. M. Quine, "A rational approximation of the Dawson’s integral for efficient computation of the complex error function", Applied Mathematics and Computation, vol. 321, pp. 526-543, 2018.
• [44] P. Scott, "Comoving coordinate system for relativistic hydrodynamics", Physical Review C, vol. 75, no. 2, pp. 1-10, 2007. doi: https://doi.org/10.1103/PhysRevC.75.024907