While going through a book named Mirror Symmetry, I came across a path integral,
$Z(\beta) = \int\limits_{X(t+\beta) = X(t)} DX(t) \exp\left(-\int\frac{1}{2}( \dot{X}^2 + X^2)dt\right)dt $
where $X(t)$ has a periodicity of $\beta$.
Using orthonormal eigenfunctions of the operator $\Theta = -\frac{d^2}{dt^2}+1 $, the exponential term is given as $\exp(-\frac{1}{2}\sum_n \lambda_n c^2_n) $, where the $\lambda_n$ are the eigenvalues. To perform the above integral, the measure is transformed as, $ DX(t) = \prod_n \frac{dc_n}{\sqrt{2\pi}}$ I didn't get how is it derived, especially the $\frac{1}{\sqrt{2\pi}}$ factor in front. How is the jacobian transformation is done here ?