Suppose that we have a function $f$ where $Q(\tau)$ is modified by multiplying $Q(\tau)$ by a real number. Let $f'$ be the modified function, and let $k \in \mathbb{R}$ or $k \in \mathbb{C}$. Take the ratio:
$\frac{f}{{f'}} = k$
$f = \exp \left( { - {{\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}\left( {\frac{\omega }{{{\omega _h}}}} \right)} }^{\frac{{ - 1}}{{\pi Q(\tau ')}}}}d\tau '} \right)\exp \left( {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} \right)\omega } d\tau '} \right)$
$f' = \exp \left( { - {{\int\limits_0^\tau {\frac{\omega }{{4Q(\tau ')}}\left( {\frac{\omega }{{{\omega _h}}}} \right)} }^{\frac{{ - 1}}{{2\pi Q(\tau ')}}}}d\tau '} \right)\exp \left( {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{2\pi Q(\tau ')}}}} - 1} \right)\omega } d\tau '} \right)$
Note that the only thing changed between $f$ and $f'$ is that $Q(\tau)$ in $f$ has become $2Q(\tau)$ in $f'$.
Is it possible to find a constant value $k$ for this ratio? For example, $k = 1/2$. Why or why not?