We present an extension of the well-known slopes method for characterization of the in-plane thermal diffusivity of semitransparent polymer films. We introduce a theoretical model which considers heat losses due to convection and radiation mechanisms, as well as semitransparency of the material to the exciting laser heat source (visible range) and multiple reflections at the film surfaces. Most importantly, a potential semitransparency of the material in the IR detection range is also considered. We prove by numerical simulations and by an asymptotic expansion of the surface temperature that the slopes method is also valid for any semitransparent film in the thermally thin regime. Measurements of the in-plane thermal diffusivity performed on semitransparent polymer films covering a wide range of absorption coefficients (to the exciting wavelength and in the IR detection range of our IR camera) validate our theoretical findings.

https://pubs.acs.org/doi/pdf/10.1021/acs.analchem.9b01583

Direct Measurement of the In-Plane Thermal Diffusivity of Semitransparent Thin Films by Lock-In Thermography: An Extension of the Slopes Method.

An algorithm is given for the numerical evaluation of a class of functions of the confluent hypergeometric type. The method of computation is based on the well-known Miller algorithms and on asymptotic expansions.

Numerical evaluation of functions arising from transformations of formal series

The paper addresses the exact evaluation of the generalized Stieltjes transform $S_{n}[f]=\int_0^{\infty} f(x) (\omega+x)^{-n}\mathrm{d}x$ of integral order $n=1,2, 3,\dots$ about $\omega =0$ from which the asymptotic behavior of $S_{n}[f]$ for small parameters $\omega$ is directly extracted. An attempt to evaluate the integral by expanding the integrand $(\omega+x)^{-n}$ about $\omega=0$ and then naively integrating the resulting infinite series term by term lead to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard's finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in [E.A. Galapon, {\it Proc. Roy. Soc. A} {\bf 473}, 20160567 (2017)]. It is shown that, when $f(x)$ does not vanish or has zero of order $m$ at the origin such that $(n-m)\geq 1$, the dominant terms of $S_{n}[f]$ as $\omega\rightarrow 0$ come from contributions arising from the poles and branch points of the complex valued function $f(z) (\omega+z)^{-n}$. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.

Finite-Part Integration of the Generalized Stieltjes Transform and its dominant asymptotic behavior for small values of the parameter Part I: Integer orders

The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f]=∫0∞f(x)(ω+x)−ndx of integral order n = 1, 2, 3, … about ω = 0 from which the asymptotic behavior of Sn[f] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x)−n about ω = 0 and then naively integrating the resulting infinite series term by term leads to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard’s finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in Galapon [Proc. R. Soc. A 473, 20160567 (2017)]. It is shown that, when f(x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f] as ω → 0 come from contributions arising from the poles and branch points of the complex valued function f(z)(ω + z)−n. These dominant terms are precisely the terms missed out by naive term by term integration. Furthermore, it is demonstrated how finite-part integration leads to new series representations of special functions by exploiting their known Stieltjes integral representations. Finally, the application of finite part integration in obtaining asymptotic expansions of the effective diffusivity in the limit of high Peclet number, the Green-Kubo formula for the self-diffusion coefficient, and the antisymmetric part of the diffusion tensor in the weak noise limit is discussed.The paper addresses the exact evaluation of the generalized Stieltjes transform Sn[f]=∫0∞f(x)(ω+x)−ndx of integral order n = 1, 2, 3, … about ω = 0 from which the asymptotic behavior of Sn[f] for small parameters ω is directly extracted. An attempt to evaluate the integral by expanding the integrand (ω + x)−n about ω = 0 and then naively integrating the resulting infinite series term by term leads to an infinite series whose terms are divergent integrals. Assigning values to the divergent integrals, say, by analytic continuation or by Hadamard’s finite part is known to reproduce only some of the correct terms of the expansion but completely misses out a group of terms. Here we evaluate explicitly the generalized Stieltjes transform by means of finite-part integration recently introduced in Galapon [Proc. R. Soc. A 473, 20160567 (2017)]. It is shown that, when f(x) does not vanish or has zero of order m at the origin such that (n − m) ≥ 1, the dominant terms of Sn[f] as ω → 0 come from contributions arisin...

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. I. Integer orders

The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied for integer n>=-1 and positive parameter q It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of q, provided special conditions on the function f(x) and its derivatives are imposed It is necessary to underline that similar formulas obtained previously are in fact asymptotic expansions only valid when q tends to infinity If one of the conditions is violated, our series become asymptotic series The validity of the formulas is illustrated by a number of examples

Exact expansions of Hankel transforms and related integrals

We obtain an exactification of the Poincare asymptotic expansion (PAE) of the Hankel integral, as , using the distributional approach of McClure & Wong. We find that, for half-integer orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the asymptotic series does not terminate and is generally divergent, but is amenable to superasymptotic summation, i.e. by optimal truncation. For specific examples, we compare the accuracy of the optimally truncated asymptotic series owing to the McClure–Wong distributional method with owing to the Mellin–Barnes integral method. We find that the former is spectacularly more accurate than the latter, by, in some cases, more than 70 orders of magnitude for the same moderate value of b. Moreover, the exactification can lead to a resummation of the PAE when it is exact, with the resummed Poincare series exhibiting again the same spectacular accuracy. More importantly, the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2013.0529

Kay Marie L. Martinez

Papers

Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales