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?
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? 


What Witten write is: Introduce a complex variable $z$ and a complexvalued polynomial $$g(z) = \sum_{j=0}^n a_n z^n.\qquad (2.7)$$ Now consider the integral $$Z_g=\intd^2z\exp\bigl(g(z)\overline{g(z)}\bigr).\qquad (2.8)$$ This again is a convergent oscillatory integral and a closer analog of complex ChernSimons 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 $$g(z)\overline{g(z)}=u+iv(uiv)=2iv$$ 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, 

