Asymptotic expansion of $(ix+a)^{q}$ as $x\rightarrow\infty$

Question

let $a>0$ be fixed and $q\in\mathbb{C}$. I would like to know the asymptotic expansion as $x\rightarrow\infty,$ $x\in\mathbb{R}$ of

$$f(x):=(ix+a)^{q}$$

This function looks so natural that I would expect this as known. For example, is there something similar to $f(x)=x^{q}(1+O(1/x))$?

Best wishes

Answer 1

As you've nearly said yourself already, this is $$i^qx^q(1+\frac{a}{ix})^q\text,$$ or, packing away some of the constant stuff to make it clearer, $$Ax^q(1+\frac{B}{x})^q\text.$$ So, apart from a constant factor of $A=i^q$, your answer looks right. Or am I missing something obvious here?