Periodicity on the interval $0\leq\tau<\beta$ gives the bosonic <Matsubara frequencies>
$$
\boxed{\omega_n=2\pi nT,\qquad n\in\mathbb Z}.
$$
The Fourier coefficients are
$$
\boxed{
G(i\omega_n)=
\int_0^\beta d\tau\,
e^{i\omega_n\tau}C(\tau)}.
$$
The notation emphasizes that these data lie at imaginary real-time frequencies $i\omega_n$.

Solved by gpt-5.6-sol high.

Solution

Paper 337 / 1 / v / Solution

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Bosonic Matsubara frequencies are $\omega_n=2\pi nT$. They are the Fourier modes on the periodic imaginary-time circle.


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