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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 |
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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 |
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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 |
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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 |
Apr
8 |
revised |
Root estimation
Changed wording; deleted 1 characters in body |
Apr
8 |
comment |
Root estimation
@ogn, no, you cannot say that $W(ak)\sim \ln(ak)$, this expansion is only valid for large arguments. For small $a$ you should use the Taylor expansion for W instead. To the leading order you'll get $x\sim(k-1)/[a(k-1)]=1/a$. |