A Small Taste from My New Book: Episode 11

 

Building on the elegant journey through asymptotic formulas and Euler’s constant in Episode 10, Episode 11 continues to illuminate the deep connections between classical analysis and special functions. In the previous episode, we explored how integral representations and analytic techniques reveal the subtle structure behind Euler’s constant and the harmonic series, uncovering the interplay between series expansions, limits, and the behavior of functions near singularities.

Now, we turn our attention to the exponential integral , a function that lies at the crossroads of analysis, number theory, and mathematical physics. Episode 11 will rigorously derive its properties, including its differentiation, series expansion, and asymptotic behavior as . We’ll see how the exponential integral encapsulates the same themes of divergence, renormalization, and analytic continuation that were central to our discussion of Euler’s constant. By dissecting its structure, we reveal how classical techniques—such as termwise integration, power series expansions, and careful handling of singularities—continue to provide powerful tools for modern mathematical analysis.

Whether you’re interested in the theoretical foundations or the practical applications, this episode offers a clear and detailed pathway to understanding the exponential integral and its role in the broader landscape of special functions. As always, the goal is to make every step transparent and accessible, ensuring that the beauty and unity of mathematical ideas shine through.

Consider the exponential integral

We’ll prove that

Differentiate

Using the form

differentiate with respect to  using the Leibniz rule for a variable lower limit:

Expand  as a series.

Recall the Taylor series of  around :

Then

So we have

as required.

In episode 10 we proved that

i.e.

From the previous step

We refine the asymptotics. For ,

so

Equivalently,

 

Integrate termwise to get

For small ,

for some constant .

Write the first few terms:

So

 

Fix the constant using the limit with .

Previously (Episode 10) we established that

That is

But from the expansion

as , the power series part tends to zero, so the limit is . Hence

So we obtain the following asymptotic expansion (small‑ expansion)

The exponential integral shows up in

• estimates for the prime counting function
• integrals involving Möbius inversion
• smoothing of sums
• the study of the Riemann zeta function via Mellin transforms

The small‑ expansion is used to isolate the logarithmic divergence and extract finite constants — exactly the same mechanism behind Euler’s constant.

In both pure and applied contexts, integrals of the form

are used to define:

• finite parts of divergent integrals
• analytic continuations
• renormalized quantities

The expansion

is the canonical decomposition into:

• the divergent part
• the finite part
• the analytic correction series

This is exactly the structure needed in renormalization schemes.

 is closely related to

• the incomplete gamma function
• the logarithmic integral
• the exponential integral

The small‑ expansion is used to derive:

• expansions of
• expansions of  near zero
• expansions of  near 1

 

Epilogue to Episode 11

In this episode, we journeyed through the intricate landscape of the exponential integral

uncovering its analytic structure, series expansions, and asymptotic behavior as . Along the way, we saw how classical techniques—termwise integration, careful handling of singularities, and the interplay between series and integrals—reveal the deep unity between analysis and special functions.

The exponential integral stands as a bridge between divergent series and finite values, encapsulating themes of renormalization and analytic continuation that echo throughout modern mathematics and physics. Its connection to Euler’s constant, the harmonic series, and the broader family of special functions highlights the enduring power of analytic methods to extract meaning from the infinite and the infinitesimal alike.

As we reflect on this exploration, it becomes clear that the true beauty of mathematics lies not only in the results, but in the clarity and rigor of the journey itself. Each step—whether differentiating under the integral sign, expanding a function near a singularity, or matching constants through careful limits—illuminates the path from classical analysis to modern mathematical thought.

If you found this episode enlightening, I encourage you to delve deeper into the detailed discussions in my books, where these techniques are developed with full transparency and step-by-step clarity. The tools and insights gained here are not just ends in themselves, but stepping stones to further discovery—reminding us that the pursuit of understanding is a journey without end.