logarithmic integral function

The logarithmic integral has an integral representation defined for all positive real numbers {{mvar|x}} ≠ 1 by the definite integral

: $\backslash operatorname\{li\}(x)\; =\; \backslash int\_0^x\; \backslash frac\{dt\}\{\backslash ln\; t\}.$

Here, {{math|ln}} denotes the natural logarithm. The function {{math|1/(ln t)}} has a singularity at {{math|1=t = 1}}, and the integral for {{math|x > 1}} is interpreted as a Cauchy principal value,

: $\backslash operatorname\{li\}(x)\; =\; \backslash lim\_\{\backslash varepsilon\; \backslash to\; 0+\}\; \backslash left(\; \backslash int\_0^\{1-\backslash varepsilon\}\; \backslash frac\{dt\}\{\backslash ln\; t\}\; +\; \backslash int\_\{1+\backslash varepsilon\}^x\; \backslash frac\{dt\}\{\backslash ln\; t\}\; \backslash right).$

However, the logarithmic integral can also be taken to be a meromorphic complex-valued function in the complex domain. In this case it is multi-valued with branch points at 0 and 1, and the values between 0 and 1 defined by the above integral are not compatible with the values beyond 1. The complex function is shown in the figure above. The values on the real axis beyond 1 are the same as defined above, but the values between 0 and 1 are offset by iπ so that the absolute value at 0 is π rather than zero. The complex function is also defined (but multi-valued) for numbers with negative real part, but on the negative real axis the values are not real.

The offset logarithmic integral or Eulerian logarithmic integral is defined as

: $\backslash operatorname\{Li\}(x)\; =\; \backslash int\_2^x\; \backslash frac\{dt\}\{\backslash ln\; t\}\; =\; \backslash operatorname\{li\}(x)\; -\; \backslash operatorname\{li\}(2).$

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

Equivalently,

: $\backslash operatorname\{li\}(x)\; =\; \backslash int\_0^x\; \backslash frac\{dt\}\{\backslash ln\; t\}\; =\; \backslash operatorname\{Li\}(x)\; +\; \backslash operatorname\{li\}(2).$

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... {{OEIS2C|A070769}}; this number is known as the Ramanujan–Soldner constant.

$\backslash operatorname\{li\}(\backslash text\{Li\}^\{-1\}(0))\; =\; \backslash text\{li\}(2)$ ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... {{OEIS2C|A069284}}

This is $-(\backslash Gamma(0,-\backslash ln\; 2)\; +\; i\backslash ,\backslash pi)$ where $\backslash Gamma(a,x)$ is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.

The function li(x) is related to the exponential integral Ei(x) via the equation

: $\backslash operatorname\{li\}(x)=\backslash hbox\{Ei\}(\backslash ln\; x)\; ,$

which is valid for x > 0. This identity provides a series representation of li(x) as

: $\backslash operatorname\{li\}(e^u)\; =\; \backslash hbox\{Ei\}(u)\; =$

\gamma + \ln |u| + \sum_{n=1}^\infty {u^{n}\over n \cdot n!}

\quad \text{ for } u \ne 0 \, ,

where γ ≈ 0.57721 56649 01532 ... {{OEIS2C|id=A001620}} is the Euler–Mascheroni constant. For the complex function the formula is

: $\backslash operatorname\{li\}(e^u)\; =\; \backslash hbox\{Ei\}(u)\; =$

\gamma + \ln u + \sum_{n=1}^\infty {u^{n}\over n \cdot n!}

\quad \text{ for } u \ne 0 \, ,

(without taking the absolute value of u).

A more rapidly convergent series by Ramanujan {{MathWorld | urlname=LogarithmicIntegral | title=Logarithmic Integral}} is

: $$

\operatorname{li}(x) =

\gamma

+ \ln |\ln x|

+ \sqrt{x} \sum_{n=1}^\infty

\left( \frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}}

\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} \right).

Again, for the meromorphic complex function the term $\backslash ln|\backslash ln\; u|$ must be replaced by $\backslash ln\backslash ln\; u.$

The asymptotic behavior both for $x\backslash to\backslash infty$ and for $x\backslash to\; 0^+$ is

: $\backslash operatorname\{li\}(x)\; =\; O\; \backslash left(\; \backslash frac\{x\; \}\{\backslash ln\; x\}\; \backslash right)\; .$

where $O$ is the big O notation. The full asymptotic expansion is

: $\backslash operatorname\{li\}(x)\; \backslash sim\; \backslash frac\{x\}\{\backslash ln\; x\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{k!\}\{(\backslash ln\; x)^k\}$

or

: $\backslash frac\{\backslash operatorname\{li\}(x)\}\{x/\backslash ln\; x\}\; \backslash sim\; 1\; +\; \backslash frac\{1\}\{\backslash ln\; x\}\; +\; \backslash frac\{2\}\{(\backslash ln\; x)^2\}\; +\; \backslash frac\{6\}\{(\backslash ln\; x)^3\}\; +\; \backslash cdots.$

This gives the following more accurate asymptotic behaviour:

: $\backslash operatorname\{li\}(x)\; -\; \backslash frac\{x\}\{\; \backslash ln\; x\}\; =\; O\; \backslash left(\; \backslash frac\{x\}\{(\backslash ln\; x)^2\}\; \backslash right)\; .$

As an asymptotic expansion, this series is not convergent: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral.

This implies e.g. that we can bracket li as:

:

for all $\backslash ln\; x\; \backslash ge\; 11$.

The logarithmic integral is important in number theory, appearing in estimates of the number of prime numbers less than a given value. For example, the prime number theorem states that:

: $\backslash pi(x)\backslash sim\backslash operatorname\{li\}(x)$

where $\backslash pi(x)$ denotes the number of primes smaller than or equal to $x$.

Assuming the Riemann hypothesis, we get the even stronger:Abramowitz and Stegun, p. 230, 5.1.20

: $|\backslash operatorname\{li\}(x)-\backslash pi(x)|\; =\; O(\backslash sqrt\{x\}\backslash log\; x)$

In fact, the Riemann hypothesis is equivalent to the statement that:

: $|\backslash operatorname\{li\}(x)-\backslash pi(x)|\; =\; O(x^\{1/2+a\})$ for any $a>0$.

For small $x$, $\backslash operatorname\{li\}(x)>\backslash pi(x)$ but the difference changes sign an infinite number of times as $x$ increases, and the first time that this happens is somewhere between 10¹⁹ and {{val|1.4|e=316}}.

• Jørgen Pedersen Gram
• Skewes' number
• List of integrals of logarithmic functions

{{reflist}}

• {{AS ref|5|228}}
• {{dlmf|id=6|title=Exponential, Logarithmic, Sine, and Cosine Integrals|first=N. M. |last=Temme}}

{{Nonelementary Integral}}

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Category:Special hypergeometric functions

Category:Integrals