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In theoretical physics, we sometimes have two perturbative expansions ofphysical quantity around different two points in parameter space. In terms ofthe two perturbative expansions, we introduce a new type of smoothinterpolating function consistent with the both expansions, which includes thestandard Pad\'e approximant and fractional power of polynomial methodconstructed by Sen as special cases. We point out that we can constructenormous number of such interpolating functions in principle while the "best"approximation for the exact answer of the physical quantity should be uniqueamong the interpolating functions. We propose a criterion to determine the"best" interpolating function, which is applicable except some situations evenif we do not know the exact answer. It turns out that our criterion works forvarious examples including specific heat in two-dimensional Ising model,average plaquette in four-dimensional SU(3) pure Yang-Mills theory on latticeand free energy in c=1 string theory at self-dual radius. We also mentionpossible applications of the interpolating functions to system with phasetransition.

On Perturbation theory improved by Strong coupling expansion