ExpIntegralE

ExpIntegralE[n,z]

gives the exponential integral function TemplateBox[{n, z}, ExpIntegralE].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{n, z}, ExpIntegralE]=int_1^inftye^(-z t)/t^ndt where the integral converges.
  • ExpIntegralE[n,z] has a branch cut discontinuity in the complex z plane running from to 0.
  • For certain special arguments, ExpIntegralE automatically evaluates to exact values.
  • ExpIntegralE can be evaluated to arbitrary numerical precision.
  • ExpIntegralE automatically threads over lists.
  • ExpIntegralE can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals for integer values of the parameter :

Plot over a subset of the complexes:

Series for generic and logarithmic cases at the origin:

Series expansion at Infinity:

Scope  (42)

Numerical Evaluation  (5)

Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Complex arguments:

Evaluate ExpIntegralE efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix ExpIntegralE function using MatrixFunction:

Specific Values  (3)

Values at fixed points:

Limiting value at infinity:

Find a real root of the equation TemplateBox[{0, x}, ExpIntegralE]=0.5:

Visualization  (3)

Plot the ExpIntegralE function:

Plot the real part of TemplateBox[{1, z}, ExpIntegralE]:

Plot the imaginary part of TemplateBox[{1, z}, ExpIntegralE]:

Plot the real part of TemplateBox[{{-, {7, /, 2}}, z}, ExpIntegralE]:

Plot the imaginary part of TemplateBox[{{-, {7, /, 2}}, z}, ExpIntegralE]:

Function Properties  (9)

Real domain of ExpIntegralE:

Complex domain of ExpIntegralE:

TemplateBox[{n, x}, ExpIntegralE] achieves all real values for :

The function range of ExpIntegralE for smaller values of may or may not be more restricted:

ExpIntegralE has the mirror property TemplateBox[{0, TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox]}, ExpIntegralE]=TemplateBox[{TemplateBox[{0, z}, ExpIntegralE]}, Conjugate]:

ExpIntegralE is not an analytic function:

Nor is it meromorphic:

TemplateBox[{n, x}, ExpIntegralE] is always nonincreasing for :

TemplateBox[{n, x}, ExpIntegralE] is injective for :

It may or may not be injective for smaller values of :

Visualize three examples:

TemplateBox[{n, x}, ExpIntegralE] is positive on :

ExpIntegralE has both singularity and discontinuity for x≤0:

TemplateBox[{n, x}, ExpIntegralE] is convex on :

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Plot higher derivatives for :

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of ExpIntegralE:

Definite integral of ExpIntegralE:

More integrals:

Series Expansions  (4)

Series expansion for ExpIntegralE:

Plot the first three approximations for TemplateBox[{1, x}, ExpIntegralE] around :

General term in the series expansion of TemplateBox[{0, x}, ExpIntegralE]:

Series expansion at infinity:

Give the result for an arbitrary symbolic direction:

ExpIntegralE can be applied to power series:

Integral Transforms  (3)

Compute the Fourier sine transform for TemplateBox[{0, t}, ExpIntegralE] using FourierSinTransform:

LaplaceTransform for TemplateBox[{1, t}, ExpIntegralE]:

MellinTransform:

Function Identities and Simplifications  (4)

Use FullSimplify to simplify exponential integrals:

Use FunctionExpand to express special cases in simpler functions:

Recurrence relationship:

For , TemplateBox[{1, x}, ExpIntegralE]=-TemplateBox[{{-, x}}, ExpIntegralEi]:

Function Representations  (5)

Primary definition of the exponential integral function:

Relationship to the incomplete gamma function Gamma:

ExpIntegralE can be represented in terms of MeijerG:

ExpIntegralE can be represented as a DifferentialRoot:

TraditionalForm formatting:

Generalizations & Extensions  (2)

Infinite arguments give exact results:

ExpIntegralE threads element-wise over lists and arrays:

Applications  (5)

Plot over the complex plane:

Solution of the heat equation for piecewise‐constant initial conditions:

Check that the solution satisfies the heat equation:

Plot the solution for different times:

Calculate a classical asymptotic series with factorial coefficients:

Plot the difference of a truncated series and the exponential integral sum:

Approximate the "leaky aquifer" function (also known as the Hantush–Jacob function or incomplete Bessel function) arising in hydrology and electronic structure calculations, using a series expansion in terms of ExpIntegralE:

Compare with quadrature of the defining integral:

Compute the expected time value of a death benefit of $1 paid at time , where is drawn from a Gompertz–Makeham distribution:

Find the annual premium, which is usually paid at the beginning of a policy year, that is necessary to make the expected time value of that payment stream for periods (where is drawn from a Gompertz–Makeham distribution) equal to the net single premium:

The resulting net annual premium:

Properties & Relations  (8)

Use FullSimplify to simplify exponential integrals:

Use FunctionExpand to express special cases in simpler functions:

Numerically find a root of a transcendental equation:

Generate from integrals, sums, and differential equations:

ExpIntegralE appears as a special case of hypergeometric and Meijer G-functions:

Integrals:

ExpIntegralE is a numeric function:

ExpIntegralE can be represented as a DifferenceRoot:

Possible Issues  (3)

Large arguments can give results too large to be computed explicitly:

Machine-number inputs can give high‐precision results:

In TraditionalForm, E_n(z) is not automatically interpreted as an exponential integral:

Neat Examples  (1)

Plot the Riemann surface of :

Wolfram Research (1988), ExpIntegralE, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpIntegralE.html (updated 2022).

Text

Wolfram Research (1988), ExpIntegralE, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpIntegralE.html (updated 2022).

CMS

Wolfram Language. 1988. "ExpIntegralE." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ExpIntegralE.html.

APA

Wolfram Language. (1988). ExpIntegralE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExpIntegralE.html

BibTeX

@misc{reference.wolfram_2025_expintegrale, author="Wolfram Research", title="{ExpIntegralE}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ExpIntegralE.html}", note=[Accessed: 05-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_expintegrale, organization={Wolfram Research}, title={ExpIntegralE}, year={2022}, url={https://reference.wolfram.com/language/ref/ExpIntegralE.html}, note=[Accessed: 05-July-2025 ]}