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
open allclose allBasic Examples (5)
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:
Evaluate ExpIntegralE efficiently at high precision:
ExpIntegralE threads elementwise over lists and matrices:
Specific Values (3)
Visualization (3)
Plot the ExpIntegralE function:
Function Properties (3)
Real domain of ExpIntegralE:
The function range of ExpIntegralE for some of the values of the argument:
Differentiation (3)
Integration (3)
Series Expansions (4)
Series expansion for ExpIntegralE:
Plot the first three approximations for around
:
General term in the series expansion of :
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 (3)
Use FullSimplify to simplify exponential integrals:
Use FunctionExpand to express special cases in simpler functions:
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)
Applications (3)
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:
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:
Text
Wolfram Research (1988), ExpIntegralE, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpIntegralE.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "ExpIntegralE." Wolfram Language & System Documentation Center. Wolfram Research. 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