A Small Taste from My New Book

 

In today’s installment of “A Small Taste from My New Book”, I want to share two contour integrals that illustrate how classical complex-analytic techniques unify a surprising amount of modern mathematics. Both integrals,

serve as small but powerful examples of how Cauchy’s integral formula, Laplace-type transforms, and analytic continuation come together in practical computations.

These two calculations touch on the central themes developed in my books “The Essential Transform Toolkit” and “The Riemann Hypothesis Revealed.”

Taken together, they illustrate the way classical contour ideas can be sharpened into versatile modern tools — whether one is extracting coefficients, evaluating generating functions, or navigating the analytic landscape behind the Riemann zeta function.

First Integral

Let’s start with the integral

where  is the vertical line defined by:

This problem is mainly connected to the inversion formula of the Laplace integral

where  is the characteristic function of , and  is the Laplace Transform of some function .

In my book The Essential Transform Toolkit, in the section on the Laplace inversion formula, I provide a detailed explanation and employ contours such as:

which makes it possible to carry out the inverse transform I begin with the vertical line in the complex plane and close the contour to the left by attaching a circular arc of radius . In this way, the contour encloses any potential singularities, allowing the application of the Residue Theorem to evaluate the integral.

In the present case, the only singularity inside the closed contour is a pole of order  located  at .

As I explain in The Essential Transform Toolkit, the contour integral reduces to the integral along the vertical line  provided the contribution from the large circular arc vanishes as . Formally this means (check out the book)

In our case, the Laplace transform is

which clearly corresponds to a simple function  (you can probably guess which one). For this transform, the condition  is obviously satisfied. So the heavy lifting is already done — the only step that remains is to compute the residue of the integrand at , which happens to be a pole of order .

To do that, expand the exponential

so

The coefficient of  (the residue at ) comes from  and equals

By the residue theorem,

Second Integral.

Claim:

where  denotes the standard keyhole contour encircling the negative real axis once counter-clockwise, and  is defined using the principal branch of the logarithm with argument in .

Put . Under this substitution, the keyhole contour   about the negative real axis in the -plane is carried to a Hankel contour  about the positive real axis in the -plane, with . 

Thus

As we traverse the contour , for  lying on it, the logarithm  can be expressed as

where .

Below, I follow the method used in the analytic continuation of the Gamma function, as presented in my book The Riemann Hypothesis Revealed (available on Amazon).

Inspect the above figure. Along the path ,  is given by , and on the path , .

Assume without loss of generality that  as , and  as . We consider  when  reaches the circular arc, i.e., .

On  we set , and on the circular path we take . Then,

and

Using the characteristic function , where  if  and  otherwise, we write the sum as:

Clearly,

and  as . Moreover, for , the integral  converges. Therefore, when , the sum simplifies to:

which leads to

We claim that

Indeed, the remaining integral, over the circular arc

has modulus less than

which vanishes as  (here, I assumed that ).

Now, since we’ve established that

the proof of the requested identity is almost complete.

In my book “The Riemann Hypothesis Revealed” the reader finds the proof and many implementations of the formula

known as Gamma functional formula or reflection formula. Using this it’s completely obvious that the asserted identity holds:

Epilogue

These two integrals are simple on the surface, yet they reveal the structural beauty of contour methods: exponentials become algebraic, coefficients become residues, and analytic continuation appears almost automatically when the contour is chosen with insight.

The techniques shown here come from two different directions — transform calculus on the one hand, and analytic number theory on the other — but they meet seamlessly through the language of complex integration. This unity is precisely what motivates the broader approaches in “The EssentialTransform Toolkit” and “The Riemann Hypothesis Revealed.”

If you found this small sample interesting, feel free to explore the full discussions in either book.

And in future posts, I’ll continue sharing short examples that highlight how far these standard complex-analytic tools can be pushed.