The sources I have been using makes it confusing for me to grasp a generalized method for the recurrence equations with root numbers.
For instance,
$T(n) = 2T(\frac{n}{4}) + \sqrt{n} $
One can write this equation in this form;
$T(n) = 2^kT(\frac{n}{2^{2k}}) + 2^{k-1}\sqrt{n} $
I cannot decide which values should I assign for the calculation of the asymptotic bounds. Frankly, I cannot continue confidently.
To elaborate my question another example might be;
$T(n) = \sqrt{n}T(\sqrt{n}) + n$
How should one treat the $n$'s ($n$ and $\sqrt{n}$) out of the recurrence equation $T(...)$ for the complexity analysis?