  We derive the Thermodynamic Bethe Ansatz (TBA) equations for the exact WKB
periods of the Mathieu equation in the weak coupling region. We will use the
TBA equations to calculate the quantum corrections to the WKB periods, which
are regarded the quantum periods of $\mathcal{N} = 2$ $SU(2)$ super Yang-Mills
theory at strong coupling. We calculate the effective central charge of the TBA
equations, which is found to be proportional to the coefficient of the one-loop
beta function of the 4d theory. We also study the spectral problem for the
Mathieu equation based on the TBA equations numerically.
