Question about parabolic cylinder functions

[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 $$for~ x<0, ~arg x^p=p\pi i)$$

 

[tiny-tim]

Hi 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 $$for~ x<0, ~arg x^p=p\pi i)$$

if x is negative, then x = |x|e^±πi

so x^p could be defined as either |x|^pe^pπi or |x|^pe^-pπi …

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

 

[PRB147]

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