# Integrals from Gradshteyn & Ryzhik: Real Part Condition Necessary?

• Asteroid
In summary, the conversation is about using an integral from Gradshteyn and Ryzhik's tables, specifically one found on page 490, for complex parameters. The participants discuss the conditions for the integral to be valid, including the real part of the parameters and avoiding the branch cut. They also explore the possibility of using the integral without the real part, but conclude that it is not recommended.
Asteroid
I have a question about an integral taken from integral tables of Gradshteyn and Ryzhik precisely , 3,914 -1 ( pag.490 ):
The condition to use the result of the integral regards the real part of the two parameters .
But if we do not have the real part but only imaginary part ,what we can do ?
I can not find an integral which permits to overcome this difficulty.
I can not understand the role of this condition, it is really necessary or in this case I can use the integral without it?

Thanks to all.

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It's pretty clear that the requirement that ##\text{Re}~\beta>0## is required for the integral to be finite. ##\text{Re}~\gamma >0## is used to avoid the branch cut that we usually take for Bessel functions on the negative real axis. You might want to check that your integral is well-defined for the range of parameters that you have.

Thanks for answer, in fact I want check the integral validity for complex parameters only without real part.
using beta e gamma as complex parameters only, i can use the result ?
I.e. since I do not have real part I can neglect the condition?
(As we do with gaussian integral, in fact this type of integral is defined for real but it can be use also for complex parameter, right?)

Let's let ##\beta = i a## and ##\gamma = i c##, then we have the integral

$$I = \int_0^\infty e^{i a \sqrt{x^2-c^2} } \cos(bx) dx.$$

I am suspicious that this could converge to a finite value, since as ##x\rightarrow \infty##, the integrand is oscillating between ##-1## and ##1##. It is clear to see that the closely related integral

$$\int_0^\infty e^{i a x } \cos(bx) dx$$

is not convergent for this reason, by direct integration and then examining the limit. On the other hand, the integral

$$\int_0^\infty e^{-x} e^{i a x } \cos(bx) dx$$

does converge and this is why that formula can be trusted when ##\text{Re}~\beta>0## (and ##\gamma## is such that the argument of the Bessel function is away from the branch cut).

I would not recommend extending the book formula to ##\text{Re}~\beta=0##.

Asteroid
Thanks for all.
It is very clear. :)

## 1. What is the purpose of the Integrals from Gradshteyn & Ryzhik book?

The Integrals from Gradshteyn & Ryzhik book is a comprehensive reference book for mathematicians, physicists, and engineers, providing a vast collection of integrals and series with detailed proofs and explanations. It is widely used as a valuable resource for solving complex integrals in various fields of science and engineering.

## 2. What types of integrals are covered in the book?

The book covers integrals and series of various types, including definite and indefinite integrals, improper integrals, multiple integrals, and special functions. It also includes integrals with complex variables and those involving rapidly convergent series.

## 3. What is the Real Part Condition for integrals?

The Real Part Condition is a necessary condition for the convergence of complex integrals. It states that if the real part of the integrand function is non-negative, then the integral will converge. This condition is essential for evaluating complex integrals using the Cauchy Integral Theorem.

## 4. How is the Real Part Condition used in evaluating integrals?

The Real Part Condition is used to simplify complex integrals by breaking them down into real and imaginary parts. By applying the condition, the integral can be transformed into a real integral, which is easier to evaluate. This approach is commonly used in the Gradshteyn & Ryzhik book to solve complex integrals.

## 5. Is the Gradshteyn & Ryzhik book suitable for beginners in mathematics?

The Gradshteyn & Ryzhik book is an advanced reference book and may not be suitable for beginners in mathematics. It assumes a strong background in calculus and complex analysis. However, it can be a helpful resource for learning about integrals and their applications once the fundamental mathematical concepts are understood.

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