Convergence of general oscillatory integral Witten, in this paper, equation (3.42), says the integral over the complex plane of the imaginary part of a polynomial g(z)
$$\int dz d\bar z  \exp( g(z)-\bar g (\bar z))$$
is convergent. How can I show it? Is there a good reference?
 A: What Witten write is:  
Introduce a complex variable $z$ and a complex-valued
polynomial
$$g(z) = \sum_{j=0}^n a_n z^n.\qquad (2.7)$$
Now consider the integral
$$Z_g=\int|d^2z|\exp\bigl(g(z)-\overline{g(z)}\bigr).\qquad  (2.8)$$
This again is a convergent oscillatory integral and a closer analog of complex
Chern-Simons theory, with $-ig(z)$ and $ig(z)$ corresponding to the terms $tW$ and 
$\widetilde{t}\overline{W}$ in the action (2.3).
We stress that no contour integral is intended in (2.8) --- a contour integral
could scarcely be intended here as the integrand is not holomorphic! Rather, 
if $z=u+iv$, with real $u$, $v$, we integrate separately over $u$ and $v$, the integration measure being $|d^2z|=2\,du\,dv$.
(end of quote)
But this is plainly false: Take $g(z)=z$, then 
$$g(z)-\overline{g(z)}=u+iv-(u-iv)=2iv$$
and the integral 
$$\int e^{2iv} du dv$$
is not convergent. 
Maybe he 
has forgotten to say that the integral is extended to a measurable set $M\subset \textbf{C}$
of finite Lebesgue measure. 
I think a more complex polynomial will not solve the problem.
Perhaps he intended to integrate only with respect to $v$. But then the measure 
will not be as he said explicitly $du dv$ but only $dv$.  The integral will not 
be absolutely convergent.  But this may be salvaged if he consider a polynomial of 
degree $\ge2$ and take principal value of the integral
Then we may translate in the following way. If $g$ is a complex plynomial of degree
$\ge2$ then 
$$\int_{-\infty}^{+\infty} e^{i\Im g(iv)} dv=\lim_{T\to+\infty}\int_{-T}^T 
e^{i\Im g(iv)} dv$$
exists. 
For example taking $g(z)=-\lambda(z^3+z)$, with $\lambda$ real,
we get $g(iv)=i\lambda( v^3/3 -v)$ so that we obtain as particular case Airy's integral
$$\int_{-\infty}^{+\infty} e^{i\lambda(v^3/3-v)} dv\qquad (2.6)$$
As the proper Witten write  in equation (2.6) of the paper.
