= Analytic subtraction of an inhomogeneous renormalization recursion
{title2=$f(u)-a(u)=b^{-D}[f(R_bu)-a(R_bu)]$}
For the <renormalization-group transformation> $f(u)=b^{-D}f(R_bu)+g(u)$, a background solving $a(u)=b^{-D}a(R_bu)+g(u)$ leaves a homogeneous remainder. In linear scaling coordinates $t'=b^{\lambda_t}t$ and $h'=b^{\lambda_h}h$, an analytic source term $g_{mn}t^mh^n$ is removed by $a_{mn}=g_{mn}/[1-b^{m\lambda_t+n\lambda_h-D}]$ unless the denominator vanishes. Such a resonance may produce a logarithm and invalidate a pure homogeneous power law. The <scaling hypothesis for critical phenomena> concerns the singular remainder. A regular field-dependent background cannot generally be represented by a function of temperature alone; an arbitrary analytic identity term proportional to $h^2$ makes this explicit.
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