For real non-zero values of x, the exponential integral Ei(x) is defined as
The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero.
For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and .[1] Instead of Ei, the following notation is used,[2]
For positive values of x, we have .
In general, a branch cut is taken on the negative real axis and E1 can be defined by analytic continuation elsewhere on the complex plane.
For positive values of the real part of , this can be written[3]
The behaviour of E1 near the branch cut can be seen by the following relation:[4]
Propertiesedit
Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.
Convergent seriesedit
For real or complex arguments off the negative real axis, can be expressed as[5]
This formula can be used to compute with floating point operations for real between 0 and 2.5. For , the result is inaccurate due to cancellation.
A faster converging series was found by Ramanujan:
Asymptotic (divergent) seriesedit
Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, more than 40 terms are required to get an answer correct to three significant figures for .[6] However, for positive values of x, there is a divergent series approximation that can be obtained by integrating by parts:[7]
The relative error of the approximation above is plotted on the figure to the right for various values of , the number of terms in the truncated sum ( in red, in pink).
Asymptotics beyond all ordersedit
Using integration by parts, we can obtain an explicit formula[8]
For any fixed , the absolute value of the error term decreases, then increases. The minimum occurs at , at which point . This bound is said to be "asymptotics beyond all orders".
Exponential and logarithmic behavior: bracketingedit
From the two series suggested in previous subsections, it follows that behaves like a negative exponential for large values of the argument and like a logarithm for small values. For positive real values of the argument, can be bracketed by elementary functions as follows:[9]
The left-hand side of this inequality is shown in the graph to the left in blue; the central part is shown in black and the right-hand side is shown in red.
Definition by Einedit
Both and can be written more simply using the entire function[10] defined as
(note that this is just the alternating series in the above definition of ). Then we have
for all z. A second solution is then given by E1(−z). In fact,
with the derivative evaluated at Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U(1,1,z):
^O’Malley, Robert E. (2014), O'Malley, Robert E. (ed.), "Asymptotic Approximations", Historical Developments in Singular Perturbations, Cham: Springer International Publishing, pp. 27–51, doi:10.1007/978-3-319-11924-3_2, ISBN 978-3-319-11924-3, retrieved 2023-05-04
^ abGiao, Pham Huy (2003-05-01). "Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution". Ground Water. 41 (3): 387–390. doi:10.1111/j.1745-6584.2003.tb02608.x. ISSN 1745-6584. PMID 12772832. S2CID 31982931.
^ abTseng, Peng-Hsiang; Lee, Tien-Chang (1998-02-26). "Numerical evaluation of exponential integral: Theis well function approximation". Journal of Hydrology. 205 (1–2): 38–51. Bibcode:1998JHyd..205...38T. doi:10.1016/S0022-1694(97)00134-0.
^Barry, D. A; Parlange, J. -Y; Li, L (2000-01-31). "Approximation for the exponential integral (Theis well function)". Journal of Hydrology. 227 (1–4): 287–291. Bibcode:2000JHyd..227..287B. doi:10.1016/S0022-1694(99)00184-5.
^"Inverse function of the Exponential Integral Ei-1(x)". Mathematics Stack Exchange. Retrieved 2024-04-24.
^George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company.
Referencesedit
Abramowitz, Milton; Irene Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Abramowitz and Stegun. New York: Dover. ISBN 978-0-486-61272-0., Chapter 5.
Bender, Carl M.; Steven A. Orszag (1978). Advanced mathematical methods for scientists and engineers. McGraw–Hill. ISBN 978-0-07-004452-4.
Bleistein, Norman; Richard A. Handelsman (1986). Asymptotic Expansions of Integrals. Dover. ISBN 978-0-486-65082-1.
Busbridge, Ida W. (1950). "On the integro-exponential function and the evaluation of some integrals involving it". Quart. J. Math. (Oxford). 1 (1): 176–184. Bibcode:1950QJMat...1..176B. doi:10.1093/qmath/1.1.176.
Stankiewicz, A. (1968). "Tables of the integro-exponential functions". Acta Astronomica. 18: 289. Bibcode:1968AcA....18..289S.
Sharma, R. R.; Zohuri, Bahman (1977). "A general method for an accurate evaluation of exponential integrals E1(x), x>0". J. Comput. Phys. 25 (2): 199–204. Bibcode:1977JCoPh..25..199S. doi:10.1016/0021-9991(77)90022-5.
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Milgram, M. S. (1985). "The generalized integro-exponential function". Mathematics of Computation. 44 (170): 443–458. doi:10.1090/S0025-5718-1985-0777276-4. JSTOR 2007964. MR 0777276.
Misra, Rama Dhar; Born, M. (1940). "On the Stability of Crystal Lattices. II". Mathematical Proceedings of the Cambridge Philosophical Society. 36 (2): 173. Bibcode:1940PCPS...36..173M. doi:10.1017/S030500410001714X. S2CID 251097063.
Chiccoli, C.; Lorenzutta, S.; Maino, G. (1988). "On the evaluation of generalized exponential integrals Eν(x)". J. Comput. Phys. 78 (2): 278–287. Bibcode:1988JCoPh..78..278C. doi:10.1016/0021-9991(88)90050-2.
Chiccoli, C.; Lorenzutta, S.; Maino, G. (1990). "Recent results for generalized exponential integrals". Computer Math. Applic. 19 (5): 21–29. doi:10.1016/0898-1221(90)90098-5.
MacLeod, Allan J. (2002). "The efficient computation of some generalised exponential integrals". J. Comput. Appl. Math. 148 (2): 363–374. doi:10.1016/S0377-0427(02)00556-3.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.3. Exponential Integrals", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, archived from the original on 2011-08-11, retrieved 2011-08-09
Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.