gives the exponential integral function TemplateBox[{z}, ExpIntegralEi].


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{z}, ExpIntegralEi]=-int_(-z)^inftye^(-t)/tdt, where the principal value of the integral is taken.
  • ExpIntegralEi[z] has a branch cut discontinuity in the complex z plane running from - to 0.
  • For certain special arguments, ExpIntegralEi automatically evaluates to exact values.
  • ExpIntegralEi can be evaluated to arbitrary numerical precision.
  • ExpIntegralEi automatically threads over lists.
  • ExpIntegralEi can be used with Interval and CenteredInterval objects. »


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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion around the branch point at the origin:

Series expansion at Infinity:

Scope  (37)

Numerical Evaluation  (5)

Evaluate numerically to high precision:

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

ExpIntegralEi can take complex number inputs:

Evaluate ExpIntegralEi efficiently at high precision:

ExpIntegralEi threads elementwise over lists and matrices:

ExpIntegralEi can be used with Interval and CenteredInterval objects:

Specific Values  (3)

Value at a fixed point:

Values at infinity:

Find the zero of the ExpIntegralEi:

Visualization  (3)

Plot the ExpIntegralEi function:

Plot the real part of TemplateBox[{z}, ExpIntegralEi]:

Plot the imaginary part of TemplateBox[{z}, ExpIntegralEi]:

Function Properties  (10)

ExpIntegralEi is defined for all real values except 0:

Complex domain:

ExpIntegralEi takes all real values:

ExpIntegralEi has the mirror property TemplateBox[{TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox]}, ExpIntegralEi]=TemplateBox[{TemplateBox[{z}, ExpIntegralEi]}, Conjugate]:

ExpIntegralEi is not an analytic function:

Nor is it meromorphic:

ExpIntegralEi is not monotonic over the reals:

However, it is monotonic over each half-line:

ExpIntegralEi is not injective:

ExpIntegralEi is surjective:

ExpIntegralEi is neither non-negative nor non-positive:

ExpIntegralEi has both singularity and discontinuity at zero:

ExpIntegralEi is neither convex nor concave:

But it is concave over the negative reals:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of ExpIntegralEi:

Definite integral of a function involving ExpIntegralEi:

More integrals:

Series Expansions  (3)

Taylor expansion for ExpIntegralEi around :

Plot the first three approximations for ExpIntegralEi around :

Find series expansion at infinity:

Give the result for an arbitrary symbolic direction:

ExpIntegralEi can be applied to power series:

Function Identities and Simplifications  (3)

Use FullSimplify to simplify expressions containing exponential integrals:

Argument simplifications:

For , TemplateBox[{x}, ExpIntegralEi]=-TemplateBox[{1, {-, x}}, ExpIntegralE]:

Function Representations  (4)

Integral representation:

ExpIntegralEi can be represented as a DifferentialRoot:

ExpIntegralEi can be represented in terms of MeijerG:

TraditionalForm formatting:

Applications  (3)

Compute a classical asymptotic series with k! coefficients:

Plot the imaginary part in the complex plane:

Real part of the EulerHeisenberg effective action:

Find a leading term in :

Properties & Relations  (8)

Use FullSimplify to simplify expressions containing exponential integrals:

Find the numerical root:

Obtain ExpIntegralEi from integrals and sums:

Calculate limits:

Obtain ExpIntegralEi from a differential equation:

Calculate Wronskian:


Integral transforms:

Possible Issues  (3)

ExpIntegralEi can take large values for moderatesize arguments:

ExpIntegralEi has a special value on the negative real axis, not obtained as a limit from either side:

A larger setting for $MaxExtraPrecision can be needed:

Neat Examples  (1)

Nested integrals:

Wolfram Research (1988), ExpIntegralEi, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpIntegralEi.html (updated 2022).


Wolfram Research (1988), ExpIntegralEi, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpIntegralEi.html (updated 2022).


Wolfram Language. 1988. "ExpIntegralEi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/ExpIntegralEi.html.


Wolfram Language. (1988). ExpIntegralEi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExpIntegralEi.html


@misc{reference.wolfram_2024_expintegralei, author="Wolfram Research", title="{ExpIntegralEi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ExpIntegralEi.html}", note=[Accessed: 20-June-2024 ]}


@online{reference.wolfram_2024_expintegralei, organization={Wolfram Research}, title={ExpIntegralEi}, year={2022}, url={https://reference.wolfram.com/language/ref/ExpIntegralEi.html}, note=[Accessed: 20-June-2024 ]}