Exponential integral

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, ${\displaystyle E_1(z)}$ can be expressed as $${\displaystyle E_1(z)=-\gamma -\ln z-\sum _{k=1}^{\mathrm{\infty }}\frac{(-z{)}^k}{kk!}(\left|\mathrm{Arg}(z)\right|<\pi )}$$ 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 ⁠${\displaystyle 0}$⁠ and ⁠${\displaystyle 2.5}$⁠. For ⁠${\displaystyle x>2.5}$⁠, the result is inaccurate due to cancellation.

A faster converging series was found by Ramanujan: $${\displaystyle \mathrm{E}\mathrm{i}(x)=\gamma +\ln x+\exp (x/2)\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n-1}{x}^n}{n!2^{n-1}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }\frac{1}{2k+1}}$$

Asymptotic (divergent) series

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 ⁠${\displaystyle E_1(10)}$⁠. However, for positive values of ⁠${\displaystyle x}$⁠, there is a divergent series approximation that can be obtained by integrating ${\displaystyle x{e}^x{E}_1(x)}$ by parts: $${\displaystyle E_1(x)=\frac{\exp (-x)}{x}\left(\sum _{n=0}^{N-1}\frac{n!}{(-x{)}^n}+O(N!x^{-N})\right)}$$ 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 $${\displaystyle \begin{array}{cc}\mathrm{Ei}(z)& =\frac{e^z}{z}\left(\sum _{k=0}^n\frac{k!}{z^k}+e_n(z)\right),\\ e_n(z)& \equiv (n+1)!z{e}^{-z}{\int }_{-\mathrm{\infty }}^z\frac{e^t}{t^{n+2}}dt.\end{array}}$$ 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 ⁠${\displaystyle |e_n(z)|\le \sqrt{\frac{2\pi }{|z|}}{e}^{-|z|}}$⁠. 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 ${\displaystyle E_1}$ 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: $${\displaystyle \frac{1}{2}{e}^{-x}\ln \left(1+\frac{2}{x}\right)<E_1(x)<e^{-x}\ln \left(1+\frac{1}{x}\right)x>0}$$

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 ${\displaystyle \mathrm{Ei}}$ and ${\displaystyle E_1}$ can be written more simply using the entire function ⁠${\displaystyle \mathrm{Ein}}$⁠ defined as $${\displaystyle \mathrm{Ein}(z)={\int }_0^z(1-e^{-t})\frac{dt}{t}=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}{z}^k}{kk!}}$$ (note that this is just the alternating series in the above definition of ⁠${\displaystyle E_1}$⁠). Then we have $${\displaystyle \begin{array}{cccc}{E}_1(z)& =-\gamma -\ln z+\mathrm{Ein}(z),& & \left|\mathrm{Arg}(z)\right|<\pi ;\\ \mathrm{Ei}(x)& =\phantom{-}\gamma +\ln x-\mathrm{E}\mathrm{i}\mathrm{n}(-x),& & x\ne 0.\end{array}}$$ The function ${\displaystyle \mathrm{Ein}}$ is related to the exponential generating function of the harmonic numbers: $${\displaystyle \mathrm{Ein}(z)=e^{-z}\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n!}{H}_n}$$

Relation with other functions

Kummer's equation $${\displaystyle z\frac{d^{2}w}{d{z}^2}+(b-z)\frac{dw}{dz}-aw=0}$$ 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, $${\displaystyle z\frac{d^{2}w}{d{z}^2}+(1-z)\frac{dw}{dz}=0}$$ we have $${\displaystyle M(0,1,z)=U(0,1,z)=1}$$ for all ⁠${\displaystyle z}$⁠. A second solution is then given by ⁠${\displaystyle E_1(-z)}$⁠. In fact, $${\displaystyle E_1(-z)=-\gamma -i\pi +\frac{\mathrm{\partial }[U(a,1,z)-M(a,1,z)]}{\mathrm{\partial }a},0<\mathrm{Arg}(z)<2\pi }$$ with the derivative evaluated at ⁠${\displaystyle a=0}$⁠. Another connexion with the confluent hypergeometric functions is that ⁠${\displaystyle E_1}$⁠ is an exponential times the function ⁠${\displaystyle U(1,1,z)}$⁠: $${\displaystyle E_1(z)=e^{-z}U(1,1,z)}$$

The exponential integral is closely related to the logarithmic integral function ⁠${\displaystyle \mathrm{li}(e^x)}$⁠ by the formula $${\displaystyle \mathrm{li}(e^x)=\mathrm{Ei}(x)}$$ for non-zero real values of ⁠${\displaystyle x}$⁠.

The series expansion of the exponential integral immediately gives rise to an expression in terms of the generalized hypergeometric function ⁠${\displaystyle {}_2{F}_2}$⁠: $${\displaystyle \mathrm{Ei}(x)=x{}_2{F}_2(1,1;2,2;x)+\ln x+\gamma .}$$

Generalization

The exponential integral may also be generalized to $${\displaystyle E_n(x)={\int }_1^{\mathrm{\infty }}\frac{e^{-xt}}{t^n}dt,}$$ which can be written as a special case of the upper incomplete gamma function: $${\displaystyle E_n(x)=x^{n-1}\mathrm{\Gamma }(1-n,x).}$$

The generalized form is sometimes called the Misra function ⁠${\displaystyle {\phi }_m(x)}$⁠, defined as $${\displaystyle {\phi }_m(x)=E_{-m}(x).}$$

Including a logarithm defines the generalized integro-exponential function $${\displaystyle E_s^j(z)=\frac{1}{\mathrm{\Gamma }(j+1)}{\int }_1^{\mathrm{\infty }}{\left(\log t\right)}^j\frac{e^{-zt}}{t^s}dt.}$$

Derivatives

The derivatives of the generalised functions ${\displaystyle E_n}$ can be calculated by means of the formula $${\displaystyle E_n^{\prime }(z)=-E_{n-1}(z)(n=1,2,3,\dots )}$$ Note that the function ${\displaystyle E_0}$ is easy to evaluate (making this recursion useful), since it is just ⁠${\displaystyle \frac{e^{-z}}{z}}$⁠.

Exponential integral of imaginary argument

If ${\displaystyle z}$ is imaginary, it has a nonnegative real part, so we can use the formula $${\displaystyle E_1(z)={\int }_1^{\mathrm{\infty }}\frac{e^{-tz}}{t}dt}$$ to get a relation with the trigonometric integrals ${\displaystyle \mathrm{Si}}$ and ⁠${\displaystyle \mathrm{Ci}}$⁠: $${\displaystyle E_1(ix)=i\left[-\frac{1}{2}\pi +\mathrm{Si}(x)\right]-\mathrm{Ci}(x)(x>0)}$$ The real and imaginary parts of ${\displaystyle E_1(ix)}$ 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 $${\displaystyle E_1(x)={\left(A^{-7.7}+B\right)}^{-0.13},}$$ where $${\displaystyle \begin{array}{cc}A& =\ln \left[\left(\frac{0.56146}{x}+0.65\right)(1+x)\right]\\ B& =x^4{e}^{7.7x}(2+x{)}^{3.7}\end{array}}$$
• The Allen and Hastings approximation $${\displaystyle E_1(x)=\{\begin{array}{ll}-\ln x+{\mathbf{\text{a}}}^{\mathsf{T}}{\mathbf{\text{x}}}_5,& x\le 1\\ \frac{e^{-x}}{x}\frac{{\mathbf{\text{b}}}^{\mathsf{T}}{\mathbf{\text{x}}}_3}{{\mathbf{\text{c}}}^{\mathsf{T}}{\mathbf{\text{x}}}_3},& x\ge 1\end{array}}$$ where $${\displaystyle \begin{array}{cc}\mathbf{\text{a}}& \triangleq [-0.57722,0.99999,-0.24991,0.05519,-0.00976,0.00108{]}^{\mathsf{T}}\\ \mathbf{\text{b}}& \triangleq [0.26777,8.63476,18.05902,8.57333{]}^{\mathsf{T}}\\ \mathbf{\text{c}}& \triangleq [3.95850,21.09965,25.63296,9.57332{]}^{\mathsf{T}}\\ {\mathbf{\text{x}}}_k& \triangleq [x^0,x^1,\dots ,x^k{]}^{\mathsf{T}}\end{array}}$$
• The continued fraction expansion $${\displaystyle E_1(x)=\frac{e^{-x}}{x+\frac{1}{1+\frac{1}{x+\frac{2}{1+\frac{2}{x+\frac{3}{\ddots }}}}}}.}$$
• The approximation of Barry et al. $${\displaystyle E_1(x)=\frac{e^{-x}}{G+(1-G)e^{-\frac{x}{1-G}}}\ln \left[1+\frac{G}{x}-\frac{1-G}{(h+bx{)}^2}\right],}$$ where: $${\displaystyle \begin{array}{cc}h& =\frac{1}{1+x\sqrt{x}}+\frac{h_{\mathrm{\infty }}q}{1+q}\\ q& =\frac{20}{47}{x}^{\sqrt{\frac{31}{26}}}\\ h_{\mathrm{\infty }}& =\frac{(1-G)(G^2-6G+12)}{3G(2-G{)}^{2}b}\\ b& =\sqrt{\frac{2(1-G)}{G(2-G)}}\\ G& =e^{-\gamma }\end{array}}$$ 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: $${\displaystyle \mathrm{\forall }|x|<\frac{\mu }{\ln (\mu )},{\mathrm{Ei}}^{-1}(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}\frac{P_n(\ln (\mu ))}{{\mu }^n}}$$ where ${\displaystyle \mu }$ is the Ramanujan–Soldner constant and ${\displaystyle (P_n)}$ is polynomial sequence defined by the following recurrence relation: $${\displaystyle P_0(x)=x,P_{n+1}(x)=x(P_n^{\prime }(x)-n{P}_n(x)).}$$

For ⁠${\displaystyle n>0}$⁠, ${\displaystyle \mathrm{deg}{P}_n=n}$ and we have the formula: $${\displaystyle P_n(x)={{\left(\frac{d}{dt}\right)}^{n-1}{\left(\frac{t{e}^x}{\mathrm{Ei}(t+x)-\mathrm{Ei}(x)}\right)}^n|}_{t=0}.}$$

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
• Solutions to the Trachenko–Zaccone nonlinear differential equation for the stretched exponential function in the relaxation of amorphous solids and glass transition