Exponential integral explained

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

\operatorname{Ei}(x) = - \int ^{inf ty} _{- x} (e ^{-t})/(tdt) = \int ^{x} _{-inf ty} (e ^{t})/(tdt.)

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]

E _{1(z)} = \int ^{inf ty} _{z} (e ^{-t})/(t) dt, |{\rm Arg}(z)|<\pi

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

z , this can be written^[3] E _{1(z)} = \int ^{inf ty} _{1} (e ^{-tz})/(t) dt = \int ^{1} _{0} (e ^{-z/u})/(u) du , \Re(z) \ge 0 .

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

\lim _{\delta\to0+} E _{1(-x} \pm i\delta) = - \operatorname{Ei}(x) \mp i\pi, 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

For real or complex arguments off the negative real axis,

E _{1(z)} can be expressed as^[5] E _{1(z)} = - \gamma - ln z - \sum ^{inf ty} _{k =1} (( - z) ^{k})/(k k!) (\left| \operatorname{Arg}(z) \right| < \pi)

where

\gamma is the Euler–Mascheroni constant. The sum converges for all complex 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

E _{1(x)} with floating point operations for real x between 0 and 2.5. For x > 2 . 5 , the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan:

{\rm Ei} (x) = \gamma + ln x + \exp{(x/ 2 )} \sum ^{inf ty} _{n =1} (( -1 ) ^{n-1} x ^{n})/(n! 2 ^{n-1}) \sum ^{\lfloor (n -1 )/ 2 \rfloor} _{k =0} (1)/(2 k +1)

Asymptotic (divergent) series

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

E _{1(10)} .^[6] However, for positive values of x, there is a divergent series approximation that can be obtained by integrating x e ^{x} E _{1(x)} by parts:^[7] E _{1 (x) = (\exp( - x))/(x)} \left(\sum ^{N -1} _{n =0} (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 N , the number of terms in the truncated sum (N =1 in red, N =5 in pink).

Asymptotics beyond all orders

Using integration by parts, we can obtain an explicit formula$$\backslash operatorname(z)\; =\; \backslash frac\; \backslash left\; (\backslash sum\; \_^\; \backslash frac\; +\; e\_(z)\backslash right),\; \backslash quad\; e\_(z)\; \backslash equiv\; (n\; +\; 1)!\backslash \; ze^\backslash int\; \_^\; \backslash frac\; \backslash ,dt$$For any fixed

z , the absolute value of the error term |e _{n(z)|} decreases, then increases. The minimum occurs at n\sim |z| , at which point \vert e _{n} (z)\vert \leq \sqrt{ (2 \pi)/(\vert z\vert) }e^. This bound is said to be "asymptotics beyond all orders".

Exponential and logarithmic behavior: bracketing

From the two series suggested in previous subsections, it follows that

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, E _{1} can be bracketed by elementary functions as follows:^[8] (1)/(2) e ^{-x} ln \left( 1+ (2)/(x) \right) < E _{1(x)} < e ^{-x} ln \left( 1+ (1)/(x) \right) x> 0

The left-hand side of this inequality is shown in the graph to the left in blue; the central part

E _{1(x)} is shown in black and the right-hand side is shown in red.

Definition by Ein

Both

\operatorname{Ei} and E _{1} can be written more simply using the entire function \operatorname{Ein} ^[9] defined as\operatorname{Ein}(z) = \int ^{z} _{0} ( 1- e ^{-t} ) (dt)/(t) = \sum ^{inf ty} _{k =1} (( -1 ) ^{k+1} z ^{k})/(k k!) (note that this is just the alternating series in the above definition of E _{1} ). Then we haveE _{1(z)} = - \gamma -ln z + {\rm Ein}(z) \left| \operatorname{Arg}(z) \right| < \pi \operatorname{Ei}(x) = \gamma +ln {x} - \operatorname{Ein}( - x) x ≠ 0

Relation with other functions

Kummer's equation

z (d ^{2w})/(dz ^{2}) + (b - z) (dw)/(dz) - aw = 0 is usually solved by the confluent hypergeometric functions M(a,b,z) and U(a,b,z). But when a =0 and b =1 , that is,z (d ^{2w})/(dz ^{2}) + ( 1- z) (dw)/(dz) = 0 we haveM( 0 , 1 ,z) = U( 0 , 1 ,z) =1 for all z. A second solution is then given by E₁(−z). In fact,E _{1 ( - z) =- \gamma - i\pi + (\partial[U(a, 1 ,z) - M(a, 1 ,z)])/(\partial a)} , 0 <{\rm Arg}(z)< 2 \pi with the derivative evaluated at a =0 . Another connexion with the confluent hypergeometric functions is that E₁ is an exponential times the function U(1,1,z):E ^{- z} _{1 (z) = e} U( 1 , 1 ,z)

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

\operatorname{li}(e ^{x)} = \operatorname{Ei}(x) for non-zero real values of x .

Generalization

The exponential integral may also be generalized to

E _{n(x)} = \int ^{inf ty} _{1} (e ^{-xt})/(t ^{n}) dt,

which can be written as a special case of the upper incomplete gamma function:^[10]

E _{n(x)} = x ^{n-1} \Gamma( 1- n,x).

The generalized form is sometimes called the Misra function^[11]

\varphi _{m(x)} , defined as\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^[12]

E ^{j(z) =} _{s} (1)/(\Gamma(j +1 )) \int ^{inf ty} _{1} \left( log t\right) ^{j} (e ^{-zt})/(t ^{s}) dt.

The indefinite integral:

\operatorname{Ei}(a ⋅ b) = \iint e ^{a} da db is similar in form to the ordinary generating function for d(n) , the number of divisors of n :\sum\limits ^{inf ty} _{n =1} d(n)x ^{n} = \sum\limits ^{inf ty} _{a =1} \sum\limits ^{inf ty} _{b =1} x ^{a}

Derivatives

The derivatives of the generalised functions

E _{n} can be calculated by means of the formula ^[13] E _{n} '(z) = - E _{n-1} (z) (n =1 , 2 , 3 ,\ldots) Note that the function E _{0} is easy to evaluate (making this recursion useful), since it is just e ^{-z} /z .^[14]

Exponential integral of imaginary argument

If

z is imaginary, it has a nonnegative real part, so we can use the formulaE _{1(z)} = \int ^{inf ty (e ^{-tz})/(t)} _{1} dt to get a relation with the trigonometric integrals \operatorname{Si} and \operatorname{Ci} :E _{1(ix)} = i\left[ - \tfrac{ 1 }{ 2 }\pi + \operatorname{Si}(x)\right] - \operatorname{Ci}(x) (x > 0 ) The real and imaginary parts of 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^[15] $$E\_1(x)\; =\; \backslash left\; (A^+B\; \backslash right)^,$$ where $$\backslash begin$$

A &= \ln\left [\left (\frac{0.56146}{x}+0.65\right)(1+x)\right] \\B &= x^4e^(2+x)^\end

• The Allen and Hastings approximation ^[16] where $$\backslash begin$$

\textbf & \triangleq [-0.57722, 0.99999, -0.24991, 0.05519, -0.00976, 0.00108]^T \\\textbf & \triangleq[0.26777,8.63476, 18.05902, 8.57333]^T \\\textbf & \triangleq[3.95850, 21.09965, 25.63296, 9.57332]^T \\\textbf_k &\triangleq[x^0,x^1,\dots, x^k]^T\end

• The continued fraction expansion $$E\_1(x)\; =\; \backslash cfrac.$$
• The approximation of Barry et al. ^[17] $$E\_1(x)\; =\; \backslash frac\backslash ln\backslash left[1+\backslash frac\; G\; x\; -\backslash frac\{1-G\}\{(h+bx)^2\}\backslash right],$$ where: $$\backslash begin$$

h &= \frac+\frac \\q &=\fracx^ \\h_ &= \frac \\b &=\sqrt \\G &= e^\end with

\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:^[18]

\forall |x| < (\mu)/(ln (\mu)) , Ei ^{-1} (x) = \sum ^{inf ty} _{n =0} (x ^{n})/(n!) (P _{n(ln(\mu))})/(\mu ^{n})

where

\mu is the Ramanujan–Soldner constant and (P _{n)} is polynomial sequence defined by the following recurrence relation:P _{0(x)} = x, P _{n+1} (x) = x(P _{n'(x)} - nP _{n(x)).}

For

n > 0 , \deg P _{n} = n and we have the formula :P _{n(x)} = \left.\left( (d)/(dt) \right) ^{n-1} \left( (te ^{x})/(Ei (t + x) - 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^[19]

See also

• Goodwin–Staton integral
• Bickley–Naylor functions

References

• Book: Abramowitz , Milton . . Irene Stegun . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Dover . 1964 . New York . 978-0-486-61272-0 . registration., Chapter 5.
• Book: Bender , Carl M. . Steven A. Orszag . Advanced mathematical methods for scientists and engineers . McGraw–Hill . 1978 . 978-0-07-004452-4.
• Book: Bleistein , Norman . Richard A. Handelsman . Asymptotic Expansions of Integrals . Dover . 1986 . 978-0-486-65082-1.
• 10.1093/qmath/1.1.176 . Ida W. . Busbridge . Quart. J. Math. (Oxford) . 1950 . 1 . 1 . On the integro-exponential function and the evaluation of some integrals involving it . 176–184 . 1950QJMat...1..176B.
• A.. Stankiewicz. Tables of the integro-exponential functions. Acta Astronomica. 18. 289. 1968. 1968AcA....18..289S.
• R. R.. Sharma. Bahman. Zohuri. A general method for an accurate evaluation of exponential integrals E₁(x), x>0. J. Comput. Phys.. 25. 2. 199–204. 10.1016/0021-9991(77)90022-5. 1977. 1977JCoPh..25..199S.
• 10.1090/S0025-5718-1983-0701632-1 . K. S.. Kölbig. On the integral exp(-μt)t^ν-1log^mt dt. Math. Comput.. 1983. 171–182. 41. 163. free.
• 10.1090/S0025-5718-1985-0777276-4 . M. S. . Milgram . Mathematics of Computation . The generalized integro-exponential function . 44 . 170 . 1985 . 0777276 . 443 - 458 . 2007964. free .
• Misra. Rama Dhar. 1940. On the Stability of Crystal Lattices. II. Mathematical Proceedings of the Cambridge Philosophical Society. 36. 2. 173. 10.1017/S030500410001714X. Born. M.. 1940PCPS...36..173M . 251097063.
• C.. Chiccoli. S.. Lorenzutta. G.. Maino. On the evaluation of generalized exponential integrals E_ν(x). J. Comput. Phys.. 78. 2. 278–287. 1988. 10.1016/0021-9991(88)90050-2. 1988JCoPh..78..278C.
• C.. Chiccoli. S.. Lorenzutta. G.. Maino. Recent results for generalized exponential integrals. Computer Math. Applic.. 19. 5. 21–29. 1990. 10.1016/0898-1221(90)90098-5.
• Allan J.. MacLeod. The efficient computation of some generalised exponential integrals. J. Comput. Appl. Math.. 10.1016/S0377-0427(02)00556-3. 2002. 148. 2. 363–374. 2002JCoAM.148..363M. free.

External links

• • NIST documentation on the Generalized Exponential Integral
    • • Exponential, Logarithmic, Sine, and Cosine Integrals in DLMF.

Notes and References

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. Bleistein and Handelsman, p. 2
7. Bleistein and Handelsman, p. 3
8. Abramowitz and Stegun, p. 229, 5.1.20
9. Abramowitz and Stegun, p. 228, see footnote 3.
10. Abramowitz and Stegun, p. 230, 5.1.45
11. After Misra (1940), p. 178
12. Milgram (1985)
13. Abramowitz and Stegun, p. 230, 5.1.26
14. Abramowitz and Stegun, p. 229, 5.1.24
15. Revisit of Well Function Approximation and An Easy Graphical Curve Matching Technique for Theis' Solution. Ground Water. 2003-05-01. 1745-6584. 387–390. 41. 3. 10.1111/j.1745-6584.2003.tb02608.x. Pham Huy. Giao. 12772832 . 2003GrWat..41..387G . 31982931 .
16. Numerical evaluation of exponential integral: Theis well function approximation. Journal of Hydrology. 1998-02-26. 38–51. 205. 1–2. 10.1016/S0022-1694(97)00134-0. Peng-Hsiang. Tseng. Tien-Chang. Lee. 1998JHyd..205...38T .
17. Approximation for the exponential integral (Theis well function) . Journal of Hydrology. 2000-01-31. 287–291. 227. 1–4. 10.1016/S0022-1694(99)00184-5. D. A. Barry. J. -Y. Parlange . L. Li. 2000JHyd..227..287B .
18. Web site: Inverse function of the Exponential Integral . 2024-04-24 . Mathematics Stack Exchange .
19. Book: Nuclear Reactor Theory. 1970. Van Nostrand Reinhold Company. George I. Bell. Samuel Glasstone.