estimate for highly oscillatory superexponential integral  I would like to estimate

$\int_{-\pi}^{\pi} e^{i n y} e^{-b e^{c y^{2}}} dy$
to within a RELATIVE error of better than 1%, if possible.  Here, $n$
is an integer and $b$ and $c$ are positive.  The imaginary component is
zero since $\sin ny$ is odd and the rest of the integrand is even.
 It is important to note that $n$ can be up to several hundred, so

this integral is very highly oscillatory.  I have tried (and am still
trying) to estimate the superexponential part of the integrand as a
piecewise polynomial, but this just gives a giant mess.
 A final note: I am trying to obtain a mathematical expression in

terms of $n$, $b$, and $c$.  There are several cases to consider (for
example, $n=0$ and $n \neq 0$), but I haven't even been able to get
the comparatively simple $n = 0$ case so far.  Any help would be
appreciated.
 A: Using Maple it is possible to obtain

When $n=0$ we have 
$$\sum _{k=0}^{\infty }{\frac {-i \left( -1 \right) ^{k}{b}^{k}\sqrt {
\pi }{{\rm erf}\left(i\sqrt {k}\sqrt {c}\pi \right)}}{k!\,\sqrt {k}
\sqrt {c}}}
$$
A: Let
$$
a_n=\int_{-\pi}^{\pi} e^{i n y}\, e^{-b e^{c y^{2}}}dy=2\int_0^{\pi} \cos(n\,y)\, e^{-b e^{c y^{2}}} dy.
$$
Let $f(y)=2\,e^{-b e^{c y^{2}}}$. Integration by parts gives
$$\begin{align}
a_n&=-\frac{1}{n}\int_0^\pi\sin(n\,y)\,f'(y)\,dy\\
&=\frac{(-1)^n}{n^2}\,f'(\pi)-\frac{1}{n^2}\int_0^\pi\cos(n\,x)\,f''(y)\,dy\\
&=\frac{(-1)^n}{n^2}\,f'(\pi)-\frac{(-1)^n}{n^4}\,f'''(\pi)+\frac{1}{n^4}\int_0^\pi\cos(n\,x)\,f^{(4)}(y)\,dy.
\end{align}$$
Then
$$
\Bigl|a_n-\frac{(-1)^n}{n^2}\,f'(\pi)\Bigr|\le\frac{C_1}{n^4}
$$
for some constant $C_1$ (that can be computed explicitly) independent of $n$. For large $n$ this should give a good approximation. If it is not enough, apply integration by parts again to obtain
$$
\Bigl|a_n-\Bigl(\frac{(-1)^n}{n^2}\,f'(\pi)-\frac{(-1)^n}{n^4}\,f'''(\pi)\Bigr)\Bigr|\le\frac{C_2}{n^6}
$$
for a constant $C_2$.
