I'm reading Kardar's book on statistical field theory. On page 52 (problem 6(f) in Chapter 3) we are asked to compute the fluctuation corrections to the mean field solution to the free energy.
In doing that this integral is given in the book as an identity
$$\int_{-\infty}^{\infty} d\phi d \phi^* \exp\left(-\beta \phi \phi^* \right) = \frac{\pi}{\beta}.$$
Here, $\phi$ and $\phi^*$ are complex-valued numbers.
I can't make sense of this identity. One can change to variables $\phi = x + i y$ and $\phi^* = x - i y$, and do the integral over the real variables $x$ and $y$. But then you end up with
$$ J \int_{-\infty}^{\infty} dx d y \exp\left(-\beta (x^2+y^2) \right) = \frac{J \pi}{\beta}.$$
Here $J$ is the Jacobian of the transformation, equal to $2i$.
Can anyone catch me up on what I'm missing? Thank you.
Indeed, thinking of $\mathbb{C}$ as a Kähler manifold, we have $$\int_{\mathbb{C}}\mathrm{e}^{-\beta z^*z}\left(\tfrac{\mathrm{i}\,\mathrm{d}z\wedge\mathrm{d}z^*}{2}\right)=\int_{\mathbb{R}^2}\mathrm{e}^{-\beta(x^2+y^2)}\mathrm{d}x\wedge\mathrm{d}y=\frac{\pi}{\beta}\text{.}$$ Statistical physicists frequently throw away constant factors when finding partition functions through integration because logarithmic derivatives of the partition function don't depend on them.