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- W4287635521 abstract "Parabolic Cylinder functions (PCFs) are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratio $Phi_n(x)=U(n-1,x)/U(n,x)$ and the double ratio $Phi_n(x)/Phi_{n+1}(x)$ in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for $U(n,z)/U(n,y)$ are also made available. The bounds are very sharp as $xrightarrow pm infty$ and $nrightarrow +infty$, and this simultaneous sharpness in three different directions explains their remarkable global accuracy. Upper and lower elementary bounds are obtained which are able to produce several digits of accuracy for moderately large $|x|$ and/or $n$." @default.
- W4287635521 created "2022-07-25" @default.
- W4287635521 creator A5060535118 @default.
- W4287635521 date "2020-10-21" @default.
- W4287635521 modified "2023-10-01" @default.
- W4287635521 title "Uniform (very) sharp bounds for ratios of Parabolic Cylinder functions" @default.
- W4287635521 doi "https://doi.org/10.48550/arxiv.2010.11118" @default.
- W4287635521 hasPublicationYear "2020" @default.
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