n th derivative of complex exponential

Question

How to find $n $ th derivative of complex exponential \begin{align*} \frac{d^{n}}{dx^{n}}e^{ix^2/(2a)} \end{align*} One method is series soultion. I want a formula which works faster in my programme. Please help me.

Answer 1

One may use Faà di Bruno's formula, with $f(x)=e^x$ and $g(x)=ix^2/2a$, whereupon we get

$$\frac{d^n}{dx^n}f(g(x))=\sum_{k=1}^n f(g(x))\cdot B_{n,k}\left(ix/2a,i/2a,0,0,\dots,0\right)$$

where $B_{n,k}$ are the bell polynomials. (closed form, though probably not good for numerical computations. As per WolframAlpha,

$$\frac{d^n}{dx^n}e^{cx^2}=2^ne^{cx^2}(cx)^nn!\sum_{k=0}^{\lfloor n/2\rfloor}\frac{(-4cx^2)^{-k}}{k!(n-2k)!}$$