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.
Examples
open allclose allBasic Examples (5)
Plot over a subset of the reals:
Series expansion at the origin:
Series expansions around the branch point at :
Series expansion at Infinity:
Scope (25)
Numerical Evaluation (4)
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:
Specific Values (4)
Visualization (2)
Function Properties (2)
LogIntegral is defined for all real positive values except 1:
LogIntegral takes all real values:
Integration (3)
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)
Applications (4)
Approximate number of primes less than :
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]:
Properties & Relations (4)
Use FullSimplify to simplify expressions into logarithmic integrals:
Use FunctionExpand to write expressions in logarithmic integrals when possible:
Obtain LogIntegral from integrals and sums:
Neat Examples (2)
Text
Wolfram Research (1988), LogIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/LogIntegral.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "LogIntegral." Wolfram Language & System Documentation Center. Wolfram Research. 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