= Compactified approximation from endpoint asymptotics
Suppose a continuous function $F(\lambda)$ on $[0,\infty)$ has a known power-law tail $F(\lambda)\sim a\lambda^{-p}$. Put $u=\lambda/(1+\lambda)$ and $h(u)=(1+\lambda)^pF(\lambda)$. When $h$ extends continuously to $u=1$, <polynomial approximation> on $[0,1]$ gives a global approximation $F(\lambda)\approx(1+\lambda)^{-p}h_N(u)$ using only elementary operations. <Taylor series> at zero and the tail <asymptotic expansion> can constrain endpoint coefficients. Agreement at the two endpoints alone does not guarantee accuracy in the intervening interval; uniform error control or intermediate calibration is needed.
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