= Solution
A finite blocking step integrates degrees of freedom on bounded length scales. Its function $g(u)$ is generally analytic near the critical couplings and contributes to a regular background in the <free-energy density>. The nonanalytic critical behavior arises from arbitrarily many blocking steps approaching the critical <renormalization-group fixed point>. After removing the analytic background, the homogeneous recursion determines the leading singular <critical exponents>, so the inhomogeneous term usually need not be retained when computing those powers.
There are qualifications. If the singular <heat capacity> has $\alpha<0$, the regular background may dominate the measured total <heat capacity>, although the singular exponent remains meaningful. At a <renormalization-group free-energy resonance>, repeated accumulation of analytic terms can generate a logarithm. For example, a term proportional to $t^2$ in $g$ has weight $b^{-jD}b^{2jy_t}$; when $D=2y_t$, every step contributes the same multiple of $t^2$. The number of steps before $t$ becomes order one is proportional to $-\ln|t|$, giving a $t^2\ln|t|$ singularity. Thus \b[the inhomogeneous term can be ignored for generic leading powers, but not indiscriminately for logarithmic corrections].
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