# ExpIntegralEi

gives the exponential integral function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• , where the principal value of the integral is taken.
• 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. »

# Examples

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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:

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 ExpIntegralEi function using MatrixFunction:

### 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 :

Plot the imaginary part of :

### Function Properties(10)

ExpIntegralEi is defined for all real values except 0:

Complex domain:

ExpIntegralEi takes all real values:

ExpIntegralEi has the mirror property :

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 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 , :

### Function Representations(4)

Integral representation:

ExpIntegralEi can be represented as a DifferentialRoot:

ExpIntegralEi can be represented in terms of MeijerG:

## 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:

Integrals:

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).

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_expintegralei, author="Wolfram Research", title="{ExpIntegralEi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/ExpIntegralEi.html}", note=[Accessed: 13-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_expintegralei, organization={Wolfram Research}, title={ExpIntegralEi}, year={2022}, url={https://reference.wolfram.com/language/ref/ExpIntegralEi.html}, note=[Accessed: 13-August-2024 ]}