# LogIntegral

LogIntegral[z]

is the logarithmic integral function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• The logarithmic integral function is defined by , where the principal value of the integral is taken.
• LogIntegral[z] has a branch cut discontinuity in the complex z plane running from to .
• For certain special arguments, LogIntegral automatically evaluates to exact values.
• LogIntegral can be evaluated to arbitrary numerical precision.
• LogIntegral automatically threads over lists.
• LogIntegral 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:

Series expansion at the origin:

Series expansions around the branch point at :

Series expansion at Infinity:

## Scope(32)

### Numerical Evaluation(5)

Evaluate numerically to high precision:

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

LogIntegral can take complex number inputs:

Evaluate LogIntegral efficiently at high precision:

LogIntegral threads elementwise over lists:

LogIntegral can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Value at the origin:

Singular point of LogIntegral:

Values at infinity:

Find a zero of :

### Visualization(2)

Plot the LogIntegral function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(8)

LogIntegral is defined for all real positive values except 1:

Complex domain:

LogIntegral takes all real values:

LogIntegral is not an analytic function:

Has both singularities and discontinuities:

LogIntegral is neither nondecreasing nor nonincreasing:

LogIntegral is not injective:

LogIntegral is surjective:

LogIntegral is neither non-negative nor non-positive:

LogIntegral is neither convex nor concave:

### Differentiation(2)

First derivative:

Higher derivatives:

### Integration(3)

Indefinite integral of LogIntegral:

Definite integral of LogIntegral:

More integrals:

### Series Expansions(3)

Taylor expansion for LogIntegral:

Plot the first three approximations for LogIntegral around :

Series expansions on either side of the branch point at :

LogIntegral can be applied to power series:

### Function Identities and Simplifications(2)

Primary definition of LogIntegral:

Use FullSimplify to simplify expressions into logarithmic integrals:

### Function Representations(3)

Representation through ExpIntegralEi:

Series representation:

TraditionalForm formatting:

## Applications(5)

Approximate number of primes less than :

Compare with exact counts:

Plot the real part in the complex plane:

Plot the absolute value in the complex plane:

Find an approximation to Soldner's constant [more info]:

for if the Riemann hypothesis is true. Verify it via LogIntegral:

## Properties & Relations(4)

Use FullSimplify to simplify expressions into logarithmic integrals:

Use FunctionExpand to write expressions in logarithmic integrals when possible:

Find the numerical root:

Obtain LogIntegral from integrals and sums:

## Possible Issues(1)

In traditional form, parentheses are needed around the argument:

## Neat Examples(2)

Nested integrals:

Plot the Riemann surface of LogIntegral:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2023_logintegral, author="Wolfram Research", title="{LogIntegral}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/LogIntegral.html}", note=[Accessed: 01-October-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_logintegral, organization={Wolfram Research}, title={LogIntegral}, year={2022}, url={https://reference.wolfram.com/language/ref/LogIntegral.html}, note=[Accessed: 01-October-2023 ]}