# ExpIntegralE

ExpIntegralE[n,z]

gives the exponential integral function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• 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 :

### Visualization(3)

Plot the ExpIntegralE function:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Real domain of ExpIntegralE:

Complex domain of 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 :

ExpIntegralE is not an analytic function:

Nor is it meromorphic:

is always nonincreasing for :

is injective for :

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

Visualize three examples:

is positive on :

ExpIntegralE has both singularity and discontinuity for x0:

is convex on :

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Plot higher derivatives for :

Formula for the 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 around :

General term in the series expansion of :

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 using FourierSinTransform:

LaplaceTransform for :

### Function Identities and Simplifications(4)

Use FullSimplify to simplify exponential integrals:

Use FunctionExpand to express special cases in simpler functions:

Recurrence relationship:

For , :

### 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:

## 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 piecewiseconstant 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 HantushJacob 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 GompertzMakeham 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 GompertzMakeham distribution) equal to the net single 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 highprecision results:

In TraditionalForm, 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_2024_expintegrale, author="Wolfram Research", title="{ExpIntegralE}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ExpIntegralE.html}", note=[Accessed: 10-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_expintegrale, organization={Wolfram Research}, title={ExpIntegralE}, year={2022}, url={https://reference.wolfram.com/language/ref/ExpIntegralE.html}, note=[Accessed: 10-September-2024 ]}