scaling of variables and parameters
(0.003 seconds)
21—27 of 27 matching pages
21: 10.43 Integrals
22: Bibliography W
23: 19.16 Definitions
24: 15.3 Graphics
25: Mathematical Introduction
26: 18.39 Applications in the Physical Sciences
§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
… βΊThe fact that both the eigenvalues of (18.39.31) and the scaling of the co-ordinate in the eigenfunctions, (18.39.30), depend on the sum leads to the substitution … βΊA major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed. … βΊwhere is a real, positive, scaling factor, and a non-negative integer. …27: Errata
In previous versions of the DLMF, in §8.18(ii), the notation was used for the scaled gamma function . Now in §8.18(ii), we adopt the notation which was introduced in Version 1.1.7 (October 15, 2022) and correspondingly, Equation (8.18.13) has been removed. In place of Equation (8.18.13), it is now mentioned to see (5.11.3).
The Olver hypergeometric function , previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint has been added. In (15.6.6), the constraint has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when , except (15.6.6) which holds for .”, has been removed.
The scaling error reported on 2016-09-12 by Dan Piponi also applied to contour and density plots for the phase of the hyperbolic umbilic canonical integrals. Scales were corrected in all figures. The interval was replaced by and replaced by . All plots and interactive visualizations were regenerated to improve image quality.
(a) Contour plot. | (b) Density plot. |
Figure 36.3.18: Phase of hyperbolic umbilic canonical integral .
(a) Contour plot. | (b) Density plot. |
Figure 36.3.19: Phase of hyperbolic umbilic canonical integral .
(a) Contour plot. | (b) Density plot. |
Figure 36.3.20: Phase of hyperbolic umbilic canonical integral .
(a) Contour plot. | (b) Density plot. |
Figure 36.3.21: Phase of hyperbolic umbilic canonical integral .
Reported 2016-09-28.
Originally the argument to in this equation was incorrect (, rather than ), and the condition on was too weak (, rather than ). Also, the factor multiplying was rewritten to clarify the poles; originally it was .
Reported 2010-11-02 by Alvaro Valenzuela.