The logarithmic integral function li(x) is defined as li (x)=P ∫0^x d t / lnt, where P indicates

step 1 We wish to evaluate this integral here, where LIX is defined as the logarithmic integral of it.

The logarithmic integral function li(x) is defined as li...

The logarithmic integral function $\mathrm{li}(x)$ is defined as li $(x)=P \int_0^x d t / \ln t$, where $P$ indicates that the Cauchy principal part of the integral is taken when $x>1$. Show that $\mathrm{li}\left(e^a\right) \sim e^a \sum_{n=0}^{\infty} n ! / a^{n+1}$ $(a \rightarrow+\infty)$

Key Concepts

Exponential Growth and Factorial Series

The asymptotic series in the problem involves terms of the form n! divided by powers of a parameter, multiplied by an exponential factor. This mix of exponential and factorial growth indicates how rapidly the function's value escalates, while the structure of the series captures successive corrections. Such series are common when approximating integrals where dominant exponential behavior is complemented by factorial corrections that refine the approximation.

Cauchy Principal Value

The Cauchy principal value is a technique used to assign a finite value to integrals that would otherwise be divergent due to a singularity in the integration domain. In the definition of the logarithmic integral, this method is indispensable because it carefully handles the singularity at t = 1 by symmetrically approaching the singular point, thereby enabling meaningful evaluation.

Logarithmic Integral Function

The logarithmic integral function, denoted li(x), is defined as a particular integral that involves the logarithmic denominator, specifically as the Cauchy principal value of ??? dt/ln(t). It appears in various areas such as number theory and plays a significant role in approximating or counting prime numbers. Handling the singularity at t = 1 using the principal value makes it a nonstandard integral requiring careful interpretation.

Asymptotic Expansion

An asymptotic expansion is a method used to approximate complex functions in a specific limit, typically when one of the variables becomes very large or very small. In this context, it allows us to represent the logarithmic integral function for large parameters as a series whose terms, involving factorials and powers, increasingly refine the approximation of the function's behavior as the parameter grows.

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