# 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:

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 :

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(2)

Compute a classical asymptotic series with k! coefficients:

Plot the imaginary part in the complex plane:

## 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: