A Small Taste from My New Book: Episode 2

 

A Small Taste from My New Book: Episode 2

Welcome to Episode 2 of "A Small Taste from My New Book." In this installment, we continue our journey through the elegant world of complex analysis, focusing on how classical contour integration techniques reveal deep connections across modern mathematics.

This episode builds on the foundational ideas introduced previously, where we explored the interplay between keyhole contours and vertical line integrals. Here, we delve deeper into the art of contour deformation method that not only simplifies challenging integrals but also illuminates the underlying unity between transform calculus and analytic number theory.

By examining how integrals over different contours can be related and transformed, we uncover the power of Cauchy's Theorem and the residue calculus. These tools allow us to transition seamlessly between representations, making it possible to evaluate integrals, extract coefficients, and even approach the analytic continuation of special functions like the Gamma and Bessel functions.

Whether you are interested in the theoretical beauty or the practical applications of these methods, this episode offers a window into the versatile toolkit that complex analysis provides.

Recap: The Core Idea of Contour Deformation

Previously [see here] I demonstrated that 

where  is the keyhole contour encircling the negative x-axis. 

In complex analysis, contour deformation is a powerful technique that allows us to change the path of integration in the complex plane, provided the integrand is analytic (holomorphic) in the region between the original and the new contour, except possibly at isolated singularities. This is justified by Cauchy's Theorem, which states that the integral of an analytic function around a closed contour is zero if there are no singularities inside the contour.

This contour can be deformed into a new , where the center of the implied semicircle is evident from the construction.

Even after this deformation, the following equality holds:

where  is a parameter implying a more general result.

This follows from applying Cauchy's Theorem to the contour , with , yielding:

As explained in TheEssential Transform Toolkit (see Example 5 in Laplace Inversion Formula) the integrals over  and  vanish.

Therefore,

The integral on the right is equal to

Indeed, substituting  for  gives  and the vertical line  becomes . The resulting integral is

which yields

Let’s now look at two straightforward examples which apply the above in practice. 

Example 1. For the integral

set :

Since  (see my book: The Reimann Hypothesis Revealed), we have:

Example 2. Clearly, we also have:

(Reader’s exercise).

 

The Magic of the Keyhole Contour

Now, let’s consider the series expansion of the Bessel function:

For a detailed explanation of this expansion, see Example 4 in the Laurent Series section of The Riemann Hypothesis Revealed.

I will deduce the following integral representations, where  denotes the keyhole contour encircling the negative x-axis.

Clearly, the integral representations come from the magic of deformation!

To prove the identity regarding the Bessel function, consider

Expand the factor  as a power series in :

Hence

Integrate

On the two rays extending away from zero, the variable  takes the form , where  and  is the radius of the circle around the origin.

We have:

­         

­         

The series’ absolute value is dominated by

For any fixed , the function

is integrable over the interval  along each ray. The exponential decay of  kills the tail as , and  avoids the singularity at . Therefore, it is justified to interchange the order of integration and summation.

On the small circle around the origin, as   the factor  blows up, while  may either suppress or amplify the integrand depending on the direction of approach. However, we don’t actually need to integrate term-by-term there. Instead, we interchange safely on the rays away from 0, then shrink the radius of the little circle to zero. In the limit, this contribution vanishes!

You can interchange the sum and the contour integral without any guilt, obtaining:

Recall that

Therefore

Compare with the Bessel series

The power-series of the Bessel function is

Thus

i.e.

Epilogue.

The examples provided, including integral representations for the Gamma and Bessel functions, demonstrate how these classical ideas not only simplify challenging computations but also reveal deep connections between transform calculus and analytic number theory. The ability to interchange summation and integration, justified by careful analysis of convergence and singularities, further underscores the elegance and practicality of these methods.

Ultimately, these techniques show that the language of complex integration serves as a bridge across diverse areas of mathematics, turning exponentials into algebraic expressions, residues into coefficients, and intricate contours into clear computational strategies. This unity is at the heart of modern mathematical thought and continues to inspire new discoveries and applications.

If you enjoyed this sample, you may wish to explore my books, 

The Riemann Hypothesis Revealed and 

The Essential Transform Toolkit, available in all Amazon marketplaces.