Q:

Is this equality between an integral and a series wrong?

In this paper (Maroun's PhD dissertation, 2013) at page 46 the following formula is given (apparently without a reference):
$$\int_0^{\infty } e^{i a x^s+i b x^p} \, dx=\sum _{n=0}^{\infty } \frac{\left(i b a^{\frac{1}{s}}\right)^n \exp \left(\frac{(i \pi ) (n p+1)}{2 s}\right) \Gamma \left(\frac{n p+1}{s}\right)}{n! a^{\frac{1}{s}} \left| s\right| }$$
Now I am trying to verify the formula. If I take $a=b=i/2$, $s=p=1$ the left hand side becomes $1$ while right hand side becomes $8/3$. Is there some error in the formula? Is this formula even well-known?

A:

We may change the variables as $xa^{1/s}=t$, this proves that $a^{1/s}\cdot \int$ is a function of $ba^{-p/s}$, not of $ba^{1/s}$.

A:

I am verifying that it is true that there is a typo. The corrected expression should be a change to the right hand side series term in parenthesis.
It should read,
$$
\left(iba^{-\frac{p}{s}}\right)^n.
$$
The original context was that the convergence is only in the sense of distributions.

