relations to hyperbolic functions
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21: 19.1 Special Notation
22: Bibliography F
23: 6.2 Definitions and Interrelations
Hyperbolic Analogs of the Sine and Cosine Integrals
… ►§6.2(iii) Auxiliary Functions
…24: 18.28 Askey–Wilson Class
§18.28(ii) Askey–Wilson Polynomials
… ►Recurrence Relation
… ►§18.28(viii) -Racah Polynomials
… ►§18.28(x) Limit Relations
… ►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).25: 19.25 Relations to Other Functions
§19.25 Relations to Other Functions
… ►§19.25(iv) Theta Functions
… ► … ►§19.25(vii) Hypergeometric Function
… ►26: 27.14 Unrestricted Partitions
§27.14(iv) Relation to Modular Functions
… ►This is related to the function in (27.14.2) by … ►The tau function is multiplicative and satisfies the more general relation: …27: Bibliography B
28: 19.2 Definitions
§19.2(iv) A Related Function:
… ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if : …The Cauchy principal value is hyperbolic: …29: 14.3 Definitions and Hypergeometric Representations
§14.3 Definitions and Hypergeometric Representations
►§14.3(i) Interval
… ►§14.3(ii) Interval
… ►§14.3(iii) Alternative Hypergeometric Representations
… ►§14.3(iv) Relations to Other Functions
…30: Errata
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) Density plot. | (b) 3D plot. |
Figure 36.3.9: Modulus of hyperbolic umbilic canonical integral function .
(a) Density plot. | (b) 3D plot. |
Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function .
(a) Density plot. | (b) 3D plot. |
Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function .
(a) Density plot. | (b) 3D plot. |
Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function .
Reported 2016-09-12 by Dan Piponi.
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.
A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.
Originally the limiting form for in the last line of this table was incorrect (, instead of ).
Reported 2010-11-23.