LogIntegral¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

LogIntegral[z]
    gives the logarithmic integral li(z), the principal value of
    Integral_0^z dt/ln t, equal to ExpIntegralEi[Log[z]], with a branch cut
    on (-Infinity, 1). LogIntegral[0] = 0, LogIntegral[1] = -Infinity,
LogIntegral[Infinity] = Infinity. Real and complex inputs evaluate
numerically at machine or arbitrary (MPFR) precision; D[LogIntegral[z], z] =
1/Log[z]. Listable.

Examples¶

No verified examples yet for this function.

Implementation notes¶

Exact special values: LogIntegral[0] = 0, LogIntegral[1] = -Infinity,

Attributes: Listable, NumericFunction, Protected.

Implementation status¶

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References¶

Source: src/info.c
Specification: docs/spec/builtins/special-functions.md

Notes & additional examples¶

Worked examples¶

In[1]:= N[LogIntegral[2], 40]
Out[1]= 1.0451637801174927848445888891946131365227

In[1]:= D[LogIntegral[z], z]
Out[1]= 1/Log[z]

In[1]:= N[LogIntegral[10^6], 30]
Out[1]= 78627.54915946218191986291074769

In[1]:= N[LogIntegral[1000], 20]
Out[1]= 177.609657990152226688

Notes¶

LogIntegral[z] is the logarithmic integral li(z), the principal value of Integral_0^z dt/Log[t], equal to ExpIntegralEi[Log[z]], with a branch cut on (-Infinity, 1). Its derivative is 1/Log[z]. li(x) is the leading term of the prime-counting approximation PrimePi[x] ~ li(x); for example li(10^6) is about 78627.5, close to PrimePi[10^6] = 78498. Real and complex inputs evaluate numerically at machine or arbitrary (MPFR) precision. LogIntegral[1] = -Infinity. Listable.