ExpIntegralE

ExpIntegralE[n,z]

gives the exponential integral function .

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 :

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 x≤0:

is convex on :

Differentiation (3)

First derivative:

Higher derivatives:

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 :

MellinTransform:

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:

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, is not automatically interpreted as an exponential integral:

Neat Examples (1)

Plot the Riemann surface of :

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