Built-in Mathematica Symbol

ExpIntegralE

ExpIntegralE[n, z]
gives the exponential integral function E_n(z).

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

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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 (3)

Evaluate numerically:

Series for generic and logarithmic cases:

Evaluate numerically:

Series for generic and logarithmic cases:

Scope (2)

Evaluate to high precision:

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

Complex arguments:

Generalizations & Extensions (3)

Infinite arguments give exact results:

ExpIntegralE threads element-wise over lists and arrays:

ExpIntegralE can be applied to power series:

Series expansion at infinity:

Give the result for an arbitrary symbolic direction:

Applications (3)

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 classic asymptotic series:

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

Properties & Relations (7)

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 special cases of hypergeometric functions:

Integrals:

ExpIntegralE is a numeric function:

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

: