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?

  • $\begingroup$ 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? $\endgroup$ Aug 30 '20 at 17:03

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