Exponential integral

Not to be confused with other integrals of exponential functions.

Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

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

For real non-zero values of x, the exponential integral Ei(x) is defined as

${\displaystyle \operatorname {Ei} (x)=-\int _{-x}^{\infty }{\frac {e^{-t}}{t}}\,dt=\int _{-\infty }^{x}{\frac {e^{t}}{t}}\,dt.}$

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 ${\displaystyle \infty }$.^[1] Instead of Ei, the following notation is used,^[2]

${\displaystyle E_{1}(z)=\int _{z}^{\infty }{\frac {e^{-t}}{t}}\,dt,\qquad |{\rm {Arg}}(z)|<\pi }$

Plot of the exponential integral function Ei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

For positive values of x, we have ${\displaystyle -E_{1}(x)=\operatorname {Ei} (-x)}$.

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 ${\displaystyle z}$, this can be written^[3]

${\displaystyle E_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,dt=\int _{0}^{1}{\frac {e^{-z/u}}{u}}\,du,\qquad \Re (z)\geq 0.}$

The behaviour of E₁ near the branch cut can be seen by the following relation:^[4]

${\displaystyle \lim _{\delta \to 0+}E_{1}(-x\pm i\delta )=-\operatorname {Ei} (x)\mp i\pi ,\qquad x>0.}$

Properties

Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.

Convergent series

Plot of ${\displaystyle E_{1}}$ function (top) and ${\displaystyle \operatorname {Ei} }$ function (bottom).

For real or complex arguments off the negative real axis, ${\displaystyle E_{1}(z)}$ can be expressed as^[5]

${\displaystyle E_{1}(z)=-\gamma -\ln z-\sum _{k=1}^{\infty }{\frac {(-z)^{k}}{k\;k!}}\qquad (\left|\operatorname {Arg} (z)\right|<\pi )}$

where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. The sum converges for all complex ${\displaystyle z}$, 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 ${\displaystyle E_{1}(x)}$ with floating point operations for real ${\displaystyle x}$ between 0 and 2.5. For ${\displaystyle x>2.5}$, the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan:^[6]

${\displaystyle {\rm {Ei}}(x)=\gamma +\ln x+\exp {(x/2)}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}x^{n}}{n!\,2^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}}$

Asymptotic (divergent) series

Relative error of the asymptotic approximation for different number ${\displaystyle ~N~}$ of terms in the truncated sum

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 ${\displaystyle E_{1}(10)}$.^[7] However, for positive values of x, there is a divergent series approximation that can be obtained by integrating ${\displaystyle xe^{x}E_{1}(x)}$ by parts:^[8]

${\displaystyle E_{1}(x)={\frac {\exp(-x)}{x}}\left(\sum _{n=0}^{N-1}{\frac {n!}{(-x)^{n}}}+O(N!x^{-N})\right)}$

The relative error of the approximation above is plotted on the figure to the right for various values of ${\displaystyle N}$, the number of terms in the truncated sum (${\displaystyle N=1}$ in red, ${\displaystyle N=5}$ in pink).

Asymptotics beyond all orders

Using integration by parts, we can obtain an explicit formula^[9]$${\displaystyle \operatorname {Ei} (z)={\frac {e^{z}}{z}}\left(\sum _{k=0}^{n}{\frac {k!}{z^{k}}}+e_{n}(z)\right),\quad e_{n}(z)\equiv (n+1)!\ ze^{-z}\int _{-\infty }^{z}{\frac {e^{t}}{t^{n+2}}}\,dt}$$For any fixed ${\displaystyle z}$, the absolute value of the error term ${\displaystyle |e_{n}(z)|}$ decreases, then increases. The minimum occurs at ${\displaystyle n\sim |z|}$, at which point ${\displaystyle \vert e_{n}(z)\vert \leq {\sqrt {\frac {2\pi }{\vert z\vert }}}e^{-\vert z\vert }}$. This bound is said to be "asymptotics beyond all orders".

Exponential and logarithmic behavior: bracketing

Bracketing of ${\displaystyle E_{1}}$ by elementary functions

From the two series suggested in previous subsections, it follows that ${\displaystyle E_{1}}$ 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, ${\displaystyle E_{1}}$ can be bracketed by elementary functions as follows:^[10]

${\displaystyle {\frac {1}{2}}e^{-x}\,\ln \!\left(1+{\frac {2}{x}}\right)<E_{1}(x)<e^{-x}\,\ln \!\left(1+{\frac {1}{x}}\right)\qquad x>0}$

The left-hand side of this inequality is shown in the graph to the left in blue; the central part ${\displaystyle E_{1}(x)}$ is shown in black and the right-hand side is shown in red.

Definition by Ein

Both ${\displaystyle \operatorname {Ei} }$ and ${\displaystyle E_{1}}$ can be written more simply using the entire function ${\displaystyle \operatorname {Ein} }$^[11] defined as

${\displaystyle \operatorname {Ein} (z)=\int _{0}^{z}(1-e^{-t}){\frac {dt}{t}}=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}z^{k}}{k\;k!}}}$

(note that this is just the alternating series in the above definition of ${\displaystyle E_{1}}$). Then we have

${\displaystyle E_{1}(z)\,=\,-\gamma -\ln z+{\rm {Ein}}(z)\qquad \left|\operatorname {Arg} (z)\right|<\pi }$

${\displaystyle \operatorname {Ei} (x)\,=\,\gamma +\ln {x}-\operatorname {Ein} (-x)\qquad x\neq 0}$

The function ${\displaystyle \operatorname {Ein} }$ is related to the exponential generating function of the harmonic numbers:

${\displaystyle \operatorname {Ein} (z)=e^{-z}\,\sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}H_{n}}$

Relation with other functions

Kummer's equation

${\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0}$

is usually solved by the confluent hypergeometric functions ${\displaystyle M(a,b,z)}$ and ${\displaystyle U(a,b,z).}$ But when ${\displaystyle a=0}$ and ${\displaystyle b=1,}$ that is,

${\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(1-z){\frac {dw}{dz}}=0}$

we have

${\displaystyle M(0,1,z)=U(0,1,z)=1}$

for all z. A second solution is then given by E₁(−z). In fact,

${\displaystyle E_{1}(-z)=-\gamma -i\pi +{\frac {\partial [U(a,1,z)-M(a,1,z)]}{\partial a}},\qquad 0<{\rm {Arg}}(z)<2\pi }$

with the derivative evaluated at ${\displaystyle a=0.}$ Another connexion with the confluent hypergeometric functions is that E₁ is an exponential times the function U(1,1,z):

${\displaystyle E_{1}(z)=e^{-z}U(1,1,z)}$

The exponential integral is closely related to the logarithmic integral function li(x) by the formula

${\displaystyle \operatorname {li} (e^{x})=\operatorname {Ei} (x)}$

for non-zero real values of ${\displaystyle x}$.

Generalization

The exponential integral may also be generalized to

${\displaystyle E_{n}(x)=\int _{1}^{\infty }{\frac {e^{-xt}}{t^{n}}}\,dt,}$

which can be written as a special case of the upper incomplete gamma function:^[12]

${\displaystyle E_{n}(x)=x^{n-1}\Gamma (1-n,x).}$

The generalized form is sometimes called the Misra function^[13] ${\displaystyle \varphi _{m}(x)}$, defined as

${\displaystyle \varphi _{m}(x)=E_{-m}(x).}$

Many properties of this generalized form can be found in the NIST Digital Library of Mathematical Functions.

Including a logarithm defines the generalized integro-exponential function^[14]

${\displaystyle E_{s}^{j}(z)={\frac {1}{\Gamma (j+1)}}\int _{1}^{\infty }\left(\log t\right)^{j}{\frac {e^{-zt}}{t^{s}}}\,dt.}$

Derivatives

The derivatives of the generalised functions ${\displaystyle E_{n}}$ can be calculated by means of the formula ^[15]

${\displaystyle E_{n}'(z)=-E_{n-1}(z)\qquad (n=1,2,3,\ldots )}$

Note that the function ${\displaystyle E_{0}}$ is easy to evaluate (making this recursion useful), since it is just ${\displaystyle e^{-z}/z}$.^[16]

Exponential integral of imaginary argument

${\displaystyle E_{1}(ix)}$ against ${\displaystyle x}$; real part black, imaginary part red.

If ${\displaystyle z}$ is imaginary, it has a nonnegative real part, so we can use the formula

${\displaystyle E_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,dt}$

to get a relation with the trigonometric integrals ${\displaystyle \operatorname {Si} }$ and ${\displaystyle \operatorname {Ci} }$:

${\displaystyle E_{1}(ix)=i\left[-{\tfrac {1}{2}}\pi +\operatorname {Si} (x)\right]-\operatorname {Ci} (x)\qquad (x>0)}$

The real and imaginary parts of ${\displaystyle \mathrm {E} _{1}(ix)}$ are plotted in the figure to the right with black and red curves.

Approximations

There have been a number of approximations for the exponential integral function. These include:

• The Swamee and Ohija approximation^[17] $${\displaystyle E_{1}(x)=\left(A^{-7.7}+B\right)^{-0.13},}$$ where $${\displaystyle {\begin{aligned}A&=\ln \left[\left({\frac {0.56146}{x}}+0.65\right)(1+x)\right]\\B&=x^{4}e^{7.7x}(2+x)^{3.7}\end{aligned}}}$$
• The Allen and Hastings approximation ^[17][18] $${\displaystyle E_{1}(x)={\begin{cases}-\ln x+{\textbf {a}}^{T}{\textbf {x}}_{5},&x\leq 1\\{\frac {e^{-x}}{x}}{\frac {{\textbf {b}}^{T}{\textbf {x}}_{3}}{{\textbf {c}}^{T}{\textbf {x}}_{3}}},&x\geq 1\end{cases}}}$$ where $${\displaystyle {\begin{aligned}{\textbf {a}}&\triangleq [-0.57722,0.99999,-0.24991,0.05519,-0.00976,0.00108]^{T}\\{\textbf {b}}&\triangleq [0.26777,8.63476,18.05902,8.57333]^{T}\\{\textbf {c}}&\triangleq [3.95850,21.09965,25.63296,9.57332]^{T}\\{\textbf {x}}_{k}&\triangleq [x^{0},x^{1},\dots ,x^{k}]^{T}\end{aligned}}}$$
• The continued fraction expansion ^[18] $${\displaystyle E_{1}(x)={\cfrac {e^{-x}}{x+{\cfrac {1}{1+{\cfrac {1}{x+{\cfrac {2}{1+{\cfrac {2}{x+{\cfrac {3}{\ddots }}}}}}}}}}}}.}$$
• The approximation of Barry et al. ^[19] $${\displaystyle E_{1}(x)={\frac {e^{-x}}{G+(1-G)e^{-{\frac {x}{1-G}}}}}\ln \left[1+{\frac {G}{x}}-{\frac {1-G}{(h+bx)^{2}}}\right],}$$ where: $${\displaystyle {\begin{aligned}h&={\frac {1}{1+x{\sqrt {x}}}}+{\frac {h_{\infty }q}{1+q}}\\q&={\frac {20}{47}}x^{\sqrt {\frac {31}{26}}}\\h_{\infty }&={\frac {(1-G)(G^{2}-6G+12)}{3G(2-G)^{2}b}}\\b&={\sqrt {\frac {2(1-G)}{G(2-G)}}}\\G&=e^{-\gamma }\end{aligned}}}$$ with ${\displaystyle \gamma }$ being the Euler–Mascheroni constant.

Inverse function of the Exponential Integral

We can express the Inverse function of the exponential integral in power series form:^[20]

${\displaystyle \forall |x|<{\frac {\mu }{\ln(\mu )}},\quad \mathrm {Ei} ^{-1}(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}{\frac {P_{n}(\ln(\mu ))}{\mu ^{n}}}}$

where ${\displaystyle \mu }$ is the Ramanujan–Soldner constant and ${\displaystyle (P_{n})}$ is polynomial sequence defined by the following recurrence relation:

${\displaystyle P_{0}(x)=x,\ P_{n+1}(x)=x(P_{n}'(x)-nP_{n}(x)).}$

For ${\displaystyle n>0}$, ${\displaystyle \deg P_{n}=n}$ and we have the formula :

${\displaystyle P_{n}(x)=\left.\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)^{n-1}\left({\frac {te^{x}}{\mathrm {Ei} (t+x)-\mathrm {Ei} (x)}}\right)^{n}\right|_{t=0}.}$

Applications

• Time-dependent heat transfer
• Nonequilibrium groundwater flow in the Theis solution (called a well function)
• Radiative transfer in stellar and planetary 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^[21]
• Solutions to the Trachenko-Zaccone nonlinear differential equation for the stretched exponential function in the relaxation of amorphous solids and glass transition^[22][23]

See also

• Goodwin–Staton integral
• Bickley–Naylor functions

Notes

1. Abramowitz and Stegun, p. 228
2. Abramowitz and Stegun, p. 228, 5.1.1
3. Abramowitz and Stegun, p. 228, 5.1.4 with n = 1
4. Abramowitz and Stegun, p. 228, 5.1.7
5. Abramowitz and Stegun, p. 229, 5.1.11
6. Andrews and Berndt, p. 130, 24.16
7. Bleistein and Handelsman, p. 2
8. Bleistein and Handelsman, p. 3
9. 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
10. Abramowitz and Stegun, p. 229, 5.1.20
11. Abramowitz and Stegun, p. 228, see footnote 3.
12. Abramowitz and Stegun, p. 230, 5.1.45
13. After Misra (1940), p. 178
14. Milgram (1985)
15. Abramowitz and Stegun, p. 230, 5.1.26
16. Abramowitz and Stegun, p. 229, 5.1.24
17. Giao, 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. Bibcode:2003GrWat..41..387G. doi:10.1111/j.1745-6584.2003.tb02608.x. ISSN 1745-6584. PMID 12772832. S2CID 31982931.
18. Tseng, 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.
19. 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.
20. "Inverse function of the Exponential Integral Ei^-1(x)". Mathematics Stack Exchange. Retrieved 2024-04-24.
21. George I. Bell; Samuel Glasstone (1970). Nuclear Reactor Theory. Van Nostrand Reinhold Company.
22. Trachenko, K.; Zaccone, A. (2021-06-14). "Slow stretched-exponential and fast compressed-exponential relaxation from local event dynamics". Journal of Physics: Condensed Matter. 33: 315101. arXiv:2010.10440. doi:10.1088/1361-648X/ac04cd. ISSN 0953-8984.
23. Ginzburg, V. V.; Gendelman, O. V.; Zaccone, A. (2024-02-23). "Unifying Physical Framework for Stretched-Exponential, Compressed-Exponential, and Logarithmic Relaxation Phenomena in Glassy Polymers". Macromolecules. 57 (5): 2520–2529. arXiv:2311.09321. doi:10.1021/acs.macromol.3c02480. ISSN 0024-9297.

References

• 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.
• Andrews, George E.; Berndt, Bruce C. (2013), Ramanujan's lost notebook. Part IV, Berlin, New York: Springer-Verlag, ISBN 978-1-4614-4080-2
• 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.
• Kölbig, K. S. (1983). "On the integral exp(−μt)t^ν−1log^mt 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. S2CID 251097063.
• Chiccoli, C.; Lorenzutta, S.; Maino, G. (1988). "On the evaluation of generalized exponential integrals E_ν(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. Bibcode:2002JCoAM.148..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, 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.

External links

• "Integral exponential function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• NIST documentation on the Generalized Exponential Integral
• Weisstein, Eric W. "Exponential Integral". MathWorld.
• Weisstein, Eric W. "En-Function". MathWorld.
• "Exponential integral Ei". Wolfram Functions Site.
• Exponential, Logarithmic, Sine, and Cosine Integrals in DLMF.