# Analyticity of determinant formula for Gaussian integral

It is a well known fact that $$\int_{\mathbb{R}^n} e^{-\frac{1}{2}x \cdot A x} dx = \sqrt{\frac{(2\pi)^n}{\det{A}}}$$ for real, positive definite $$A$$. The left hand side of the equation make sense for any complex-symmetric $$A$$ with real part positive definite. The left hand side is also analytic in $$A$$ under this assumption of absolute convergence. The right hand side is analytic in $$A$$ except for a potential branch cut of the square root function.

However I'm unsure under what conditions one can analytically continue this formula. I have produced a counterexample where $$A$$ is complex symmetric and $$\Re{A}$$ is positive definite but yet the formula is wrong by a phase, e.g. take $$A = e^{i\phi} \text{Id}$$ and let $$\frac{\pi}{n} < |\phi|< \frac{\pi}{2}$$ (obviously we need $$n \geq 3$$ for this example). Under what circumstances is the formula valid?

• Maybe it’s easier to figure this out if you square both sides and figure out where that is valid, then focus on taking the square root? Aug 30 '20 at 17:03