In mathematics, theexponential integral Ei is aspecial function on thecomplex plane.

It is defined as one particulardefinite integral of the ratio between anexponential function and itsargument.

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

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

TheRisch algorithm shows that Ei is not anelementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of theCauchy principal value due to the singularity of the integrand at zero.

For complex values of the argument, the definition becomes ambiguous due tobranch points at 0 and${\displaystyle \mathrm{\infty }}$.^[1] Instead of Ei, the following notation is used,^[2]

For positive values of x, we have${\displaystyle -E_1(x)=\mathrm{Ei}(-x)}$.

In general, abranch cut is taken on the negative real axis andE₁ can be defined byanalytic continuation elsewhere on the complex plane.

For positive values of the real part of${\displaystyle z}$, this can be written^[3]

${\displaystyle E_1(z)={\int }_1^{\mathrm{\infty }}\frac{e^{-tz}}{t}dt={\int }_0^1\frac{e^{-z/u}}{u}du,\mathrm{\Re }(z)\ge 0.}$

The behaviour ofE₁ near the branch cut can be seen by the following relation:^[4]

${\displaystyle \underset{\delta \to 0+}{lim}{E}_1(-x\pm i\delta )=-\mathrm{Ei}(x)\mp i\pi ,x>0.}$

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

For real or complex arguments off the negative real axis,${\displaystyle E_1(z)}$ can be expressed as^[5]

where${\displaystyle \gamma }$ is theEuler–Mascheroni constant. The sum converges for all complex${\displaystyle z}$, and we take the usual value of thecomplex logarithm having abranch 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 tocancellation.

A faster converging series was found byRamanujan:^[6]

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

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 threesignificant 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${\displaystyle x{e}^x{E}_1(x)}$ 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).

Usingintegration by parts, we can obtain an explicit formula^[9]$${\displaystyle \mathrm{Ei}(z)=\frac{e^z}{z}\left(\sum _{k=0}^n\frac{k!}{z^k}+e_n(z)\right),e_n(z)\equiv (n+1)!z{e}^{-z}{\int }_{-\mathrm{\infty }}^z\frac{e^t}{t^{n+2}}dt}$$ 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".

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]

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.

Both${\displaystyle \mathrm{Ei}}$ and${\displaystyle E_1}$ can be written more simply using theentire function${\displaystyle \mathrm{Ein}}$^[11] 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 thealternating series in the above definition of${\displaystyle E_1}$). Then we have

${\displaystyle \mathrm{Ei}(x)=\gamma +\ln x-\mathrm{Ein}(-x)x\ne 0}$

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

${\displaystyle \mathrm{Ein}(z)=e^{-z}\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n!}{H}_n}$

Kummer's equation

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

is usually solved by theconfluent 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 allz. A second solution is then given by E₁(−z). In fact,

with the derivative evaluated at${\displaystyle a=0.}$ Another connexion with the confluent hypergeometric functions is thatE₁ is an exponential times the functionU(1,1,z):

${\displaystyle E_1(z)=e^{-z}U(1,1,z)}$

The exponential integral is closely related to thelogarithmic integral function li(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${\displaystyle {}_2{F}_2}$:

${\displaystyle \mathrm{Ei}(x)=x{}_2{F}_2(1,1;2,2;x)+\ln x+\gamma .}$

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 upperincomplete gamma function:^[12]

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

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

${\displaystyle {\phi }_m(x)=E_{-m}(x).}$

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

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

${\displaystyle E_s^j(z)=\frac{1}{\mathrm{\Gamma }(j+1)}{\int }_1^{\mathrm{\infty }}{\left(\log t\right)}^j\frac{e^{-zt}}{t^s}dt.}$

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

${\displaystyle E_n^{\prime }(z)=-E_{n-1}(z)(n=1,2,3,\dots )}$

Note that the function${\displaystyle E_0}$ is easy to evaluate (making this recursion useful), since it is just${\displaystyle e^{-z}/z}$.^[16]

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 thetrigonometric 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${\displaystyle {\mathrm{E}}_1(ix)}$ are plotted in the figure to the right with black and red curves.

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
• The Allen and Hastings approximation^[17][18] where
• The continued fraction expansion^[18]$${\displaystyle E_1(x)=\frac{e^{-x}}{x+\frac{1}{1+\frac{1}{x+\frac{2}{1+\frac{2}{x+\frac{3}{\ddots }}}}}}.}$$
• The approximation of Barryet 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: with${\displaystyle \gamma }$ being theEuler–Mascheroni constant.

We can express theInverse function of the exponential integral inpower series form:^[20]

where${\displaystyle \mu }$ is theRamanujan–Soldner constant and${\displaystyle (P_n)}$ ispolynomial sequence defined by the followingrecurrence 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 :

${\displaystyle P_n(x)={{\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)}^{n-1}{\left(\frac{t{e}^x}{\mathrm{E}\mathrm{i}(t+x)-\mathrm{E}\mathrm{i}(x)}\right)}^n|}_{t=0}.}$

• Time-dependentheat transfer
• Nonequilibriumgroundwater flow in theTheis solution (called awell function)
• Radiative transfer in stellar and planetary atmospheres
• Radial diffusivity equation for transient or unsteady state flow with line sources and sinks
• Solutions to theneutron transport equation in simplified 1-D geometries^[21]
• Solutions to theTrachenko-Zacconenonlinear differential equation for thestretched exponential function in the relaxation ofamorphous solids andglass transition^[22][23]

• Goodwin–Staton integral
• Bickley–Naylor functions

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• Special hypergeometric functions
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