is the logarithmic integral function .


  • 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.


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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  (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)

Value at the origin:

Singular point of LogIntegral:

Values at infinity:

Find a zero of TemplateBox[{x}, LogIntegral]:

Visualization  (2)

Plot the LogIntegral function:

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}}, LogIntegral]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, LogIntegral]:

Function Properties  (2)

LogIntegral is defined for all real positive values except 1:

Complex domain:

LogIntegral takes all real values:

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  (4)

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]:

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:

Introduced in 1988