Aleksey Pichugin
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 Jan 25 awarded Enlightened Jan 25 awarded Nice Answer Jun 4 comment Rigorous results on the method of multiple scales However, be warned: these monographs are mainly focussed on "classic" homogenization, in which the leading order problem is, essentially, one-scale. The extensions of these techniques to problems that are essentially two-scale even at the leading order were developed by G. Nguetseng and G. Allaire (the paper by G. Allaire called "Homogenization and two-scale convergence", SIAM J. Math. Anal., 23(1992), pp. 1482-1518 is a popular entry point to this area). Jun 4 comment Rigorous results on the method of multiple scales A number of results similar to what you are looking for were obtained in the literature on asymptotic homogenization. The classic monographs in these area are: Bensoussan, Lions & Papanicolaou (1978) "Asymptotic analysis for periodic structures", Sanchez-Palencia (1980) "Non-homogeneous media and vibration theory" and Bakhvalov & Panasenko (1989) "Homogenization: averaging processes in periodic media". They all use the method of multiple scales to get a formal expansion and then derive rigorous estimates on the accuracy of the resulting asymptotics. Jun 4 comment Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? @Robert Israel: Thanks you very much. This also works in Maple 13. Jun 3 comment Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? Just found a formula underlying Robert Israel's result in Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10): \begin{align} & s_{\mu,\nu}(z)=\frac{z^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)} \\ &\qquad\qquad \times{}_1F_2\left( {\textstyle 1;\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{3}{2},\frac{1}{2}\mu+\frac{1}{2}\nu+\frac{‌​3}{2};-\frac{1}{4}z^2} \right)\,. \end{align} Jun 3 comment Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? @Robert Israel: This is very interesting. Could you please explain how you obtained this identity? I use Maple 13 and have not been able to obtain something this compact directly. Jun 3 comment Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? This is a proper link to the mentioned paper: F.J.W. Whipple (1927) "", J. Lond. Math. Soc., **2**(2), pp. 85-90. Jun 3 awarded Student Jun 2 revised Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? Corrected spelling Jun 2 asked Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? Nov 4 comment Definite Integral ∫_{0}^{∞} dx exp(−x^2−a exp(b x^2)) Is it the accuracy or qualitative understanding you are looking for here? Have you considered deriving asymptotic solutions? Large $a$ expansion comes to mind first, but other things must also be possible. Nov 3 revised Analytical solutions of a differential equation (from Archimedes' Spiral) Added explicit expression for coefficients of the McLaurin series expansion and clarified the accuracy of the Pade approximants Nov 1 answered Analytical solutions of a differential equation (from Archimedes' Spiral) Aug 21 awarded Yearling Apr 9 revised Root estimation Added recurrent relationship that enables arbitrary precision Apr 8 comment Root estimation This is a very efficient method of generating formal power series, thank you for sharing. I was so fixed on the expansion of first type, could not see the other option. Just a small comment: could you please replace the sum variable: should be $n$, not $k$ (I was struggling with this notation too!). Apr 8 revised Root estimation Modified eq.(3) to show its relation to eq.(2) Apr 8 comment Root estimation @ogn, I did not try to argue that Zander's approximation is uniform, I only commented on your estimate. Apr 8 awarded Commentator