# Question about parabolic cylinder functions

• PRB147
So, when x<0, x^p would have a negative imaginary part, which is equal to pπi. This is just a notation to indicate the branch cut for the complex logarithm. In summary, the expression D_p(z)=\int_{-\infty}^{\infty}x^p e^{-2x^2+2i xz}dx represents an integral with the condition Re~ p>-1, and for x<0, the argument of x^p is equal to pπi due to the branch cut for the complex logarithm.
PRB147
In table of integrals, series and products 7ed. by Gradshtyn and Ryzhik,
in page 1028, there is an expression:
$$D_p(z)=\int_{-\infty}^{\infty}x^p e^{-2x^2+2i xz}dx,~~(Re~ p>-1; ~for~ x<0, ~arg x^p=p\pi i)$$

what is the meaning of $$for~ x<0, ~arg x^p=p\pi i)$$

Last edited:
Hi PRB147!
PRB147 said:
In table of integrals, series and products 7ed. by Gradshtyn and Ryzhik,
in page 1028, there is an expression:
$$D_p(z)=\int_{-\infty}^{\infty}x^p e^{-2x^2+2i xz}dx,~~(Re~ p>-1; ~for~ x<0, ~arg x^p=p\pi i)$$

what is the meaning of $$for~ x<0, ~arg x^p=p\pi i)$$

if x is negative, then x = |x|e±πi

so xp could be defined as either |x|pepπi or |x|pe-pπi

the question is merely telling you to adopt the former definition!

thank you, tiny-tim!
I remember that arg(z) is a real number.

## What are parabolic cylinder functions?

Parabolic cylinder functions, also known as Weber functions, are special functions in mathematics that are solutions to a second-order differential equation known as the parabolic cylinder equation. They are closely related to the Gaussian function and play an important role in various fields of science, including physics, engineering, and statistics.

## What is the difference between parabolic cylinder functions and other special functions?

Unlike many other special functions, parabolic cylinder functions are not typically expressed in terms of elementary functions such as polynomials, trigonometric functions, or exponential functions. Instead, they are defined as solutions to a specific differential equation. This makes them more difficult to manipulate and work with, but also makes them useful in solving complex mathematical problems.

## What are the applications of parabolic cylinder functions?

Parabolic cylinder functions have a wide range of applications in various fields of science and engineering. They are commonly used in physics to model the behavior of quantum particles, in electrical engineering to describe the propagation of electromagnetic waves, and in statistics to analyze data and make predictions.

## How are parabolic cylinder functions related to other special functions?

Parabolic cylinder functions have close connections to many other special functions, including Hermite polynomials, Bessel functions, and hypergeometric functions. They also have connections to other mathematical concepts, such as orthogonal polynomials and confluent hypergeometric functions.

## How can I use parabolic cylinder functions in my research or work?

If you are a scientist or researcher, you may encounter situations where parabolic cylinder functions can be used to solve problems or model phenomena in your field. It is important to have a good understanding of these functions and how they can be applied in order to effectively use them in your work. You may also consult with a mathematician or other experts in your field for guidance on using parabolic cylinder functions in your research.

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