ExpIntegralE

ExpIntegralE[n,z]

gives the exponential integral function .

Details

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

Examples

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Basic Examples  (5)

Evaluate numerically:

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

Plot over a subset of the complexes:

Series for generic and logarithmic cases at the origin:

Series expansion at Infinity:

Scope  (34)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

Complex arguments:

Evaluate ExpIntegralE efficiently at high precision:

ExpIntegralE threads elementwise over lists and matrices:

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, {x, +, {ⅈ,  , y}}}, ExpIntegralE]:

Plot the imaginary part of TemplateBox[{1, {x, +, {ⅈ,  , y}}}, ExpIntegralE]:

Plot the real part of TemplateBox[{{-, {7, /, 2}}, {x, +, {ⅈ,  , y}}}, ExpIntegralE]:

Plot the imaginary part of TemplateBox[{{-, {7, /, 2}}, {x, +, {ⅈ,  , y}}}, ExpIntegralE]:

Function Properties  (3)

Real domain of ExpIntegralE:

Complex domain:

The function range of ExpIntegralE for some of the values of the argument:

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

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  (3)

Use FullSimplify to simplify exponential integrals:

Use FunctionExpand to express special cases in simpler functions:

Recurrence relationship:

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  (3)

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 classic asymptotic series:

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

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

Neat Examples  (1)

Plot the Riemann surface of :

Introduced in 1988
 (1.0)