function (top) and Ei function (bottom).
In mathematics, the
exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions [edit ]
For real nonzero values of
x, the exponential integral Ei( x) is defined as
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 . Instead of Ei, the following notation is used, [1 ] [2 ]
In general, a
branch cut is taken on the negative real axis and E 1 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 E
1 near the branch cut can be seen by the following relation: [4 ] Properties [edit ]
Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.
Convergent series [edit ]
Taylor series for , and extracting the logarithmic singularity, we can derive the following series representation for for real : [5 ]
For complex arguments off the negative real axis, this generalises to
is the Euler–Mascheroni constant. The sum converges for all complex , and we take the usual value of the complex logarithm having a branch cut along the negative real axis.
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) series [edit ]
Relative error of the asymptotic approximation for different number
of terms in the truncated sum
Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, for x=10 more than 40 terms are required to get an answer correct to three significant figures.
However, there is a divergent series approximation that can be obtained by integrating [7 ] by parts: [8 ]
which has error of order
and is valid for large values of . 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). Exponential and logarithmic behavior: bracketing [edit ]
by elementary functions
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 [edit ]
and can be written more simply using the entire function defined as [10 ]
(note that this is just the alternating series in the above definition of
). Then we have Relation with other functions [edit ]
The exponential integral is closely related to the
logarithmic integral function li( x) by the formula
for positive real values of
The exponential integral may also be generalized to
which can be written as a special case of the
incomplete gamma function: [11 ]
The generalized form is sometimes called the Misra function
[12 ] , defined as
Including a logarithm defines the generalized integro-exponential function
[13 ] .
The indefinite integral:
is similar in form to the ordinary
generating function for , the number of divisors of : Derivatives [edit ]
The derivatives of the generalised functions
can be calculated by means of the formula [14 ]
Note that the function
is easy to evaluate (making this recursion useful), since it is just . [15 ] Exponential integral of imaginary argument [edit ]
; real part black, imaginary part red.
is imaginary, it has a nonnegative real part, so we can use the formula
to get a relation with the
trigonometric integrals and :
The real and imaginary parts of
are plotted in the figure to the right with black and red curves. Applications [edit ] Time-dependent heat transfer Nonequilibrium groundwater flow in the Theis solution (called a well function) Radiative transfer in stellar atmospheres Radial diffusivity equation for transient or unsteady state flow with line sources and sinks Solutions to the neutron transport equation in simplified 1-D geometries. [16 ] Notes [edit ] ^ Abramowitz and Stegun, p. 228 ^ Abramowitz and Stegun, p. 228, 5.1.1 ^ Abramowitz and Stegun, p. 228, 5.1.4 with n = 1 ^ Abramowitz and Stegun, p. 228, 5.1.7 ^ For a derivation, see Bender and Orszag, p253 ^ Abramowitz and Stegun, p. 229, 5.1.11 ^ Bleistein and Handelsman, p. 2 ^ Bleistein and Handelsman, p. 3 ^ Abramowitz and Stegun, p. 229, 5.1.20 ^ Abramowitz and Stegun, p. 228, see footnote 3. ^ Abramowitz and Stegun, p. 230, 5.1.45 ^ After Misra (1940), p. 178 ^ Milgram (1985) ^ Abramowitz and Stegun, p. 230, 5.1.26 ^ Abramowitz and Stegun, p. 229, 5.1.24 ^ George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company. References [edit ] Abramovitz, Milton; Irene Stegun (1964). . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables Abramowitz and Stegun. New York: Dover. ISBN 0-486-61272-4. , Chapter 5. Bender, Carl M.; Steven A. Orszag (1978). Advanced mathematical methods for scientists and engineers. McGraw–Hill. ISBN 0-07-004452-X. Bleistein, Norman; Richard A. Handelsman (1986). Asymptotic Expansions of Integrals. Dover. ISBN 0-486-65082-0. 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 E 1(x), x>0". J. Comput. Phys. 25 (2): 199—204. Bibcode: 1977JCoPh..25..199S. doi: 10.1016/0021-9991(77)90022-5. Kölbig, K. S. (1983). "On the integral exp(− μt) t ν−1log m t dt". Math. Comput 41 (163): 171—182. doi: 10.1090/S0025-5718-1983-0701632-1. 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. Chiccoli, C.; Lorenzutta, S.; Maino, G. (1988). "On the evaluation of generalized exponential integrals E ν(x)". J. Comput. Phys. 78: 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. Bibcode: 2002JCoAm.138..363M. 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 Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., , Cambridge University Press, NIST Handbook of Mathematical Functions ISBN 978-0521192255, MR 2723248 External links [edit ] Hazewinkel, Michiel, ed. (2001), "Integral exponential function", , Encyclopedia of Mathematics Springer, ISBN 978-1-55608-010-4 NIST documentation on the Generalized Exponential Integral Weisstein, Eric W., " Exponential Integral", . MathWorld Weisstein, Eric W., " ", En-Function . MathWorld Formulas and identities for Ei