Magnus Expansion in Floquet theory [closed]

Question

I wonder how to obtain the second equality as follows in Eq. (44) of

Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. M Bukov, L D'Alessio & A Polkovnikov. Adv. Phys. 64, 139 (2015), arXiv:1407.4803.

which reads

\begin{eqnarray} K_\mathrm{eff}^{(1)}[t_0](t)&=&\frac{1}{\hbar}\int_{t_0}^t dt'(H(t')-H_F^{(1)}[t_0]) \nonumber \\ &=& -\frac{1}{2\hbar} \left[\int_{t}^{t+T} dt'H(t') \left(1+2\frac{t-t'}{T} \right)-\int_{t_0}^{t_0+T} dt'H(t') \left(1+2\frac{t_0-t'}{T} \right) \right] . \end{eqnarray}

Answer 1

One way to see this is using the last equality of Eq.(48), i.e.

$K_F^{(1)}[t_0](t) = K_\mathrm{eff}^{(1)}(t) - K_\mathrm{eff}^{(1)}(t_0)$

and then applying Eq.(47). A bit confusing given that Eq.(44) appears in the previous subsection.

Also, note that the weight function

$f(t-t') = (1+2\frac{t-t'}{T})$

appearing in the integrand of Eq.(47) [and thus also Eq.(44)] is defined periodic in $[0,T]$ with period $T$, i.e. $f(x+T) = f(x)$. So be careful if you need to shift the integration variable $t'$ ;)