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11: 36.7 Zeros
§36.7(iii) Elliptic Umbilic Canonical Integral
… ►The zeros are lines in space where is undetermined. …Near , and for small and , the modulus has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose and repeat distances are given by … ►§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals
►The zeros of these functions are curves in space; see Nye (2007) for and Nye (2006) for .12: 20 Theta Functions
Chapter 20 Theta Functions
…13: Bibliography B
14: Bibliography U
15: 4.35 Identities
§4.35 Identities
►§4.35(i) Addition Formulas
… ►§4.35(ii) Squares and Products
… ►§4.35(iii) Multiples of the Argument
… ►§4.35(iv) Real and Imaginary Parts; Moduli
…16: 4.29 Graphics
§4.29(i) Real Arguments
… ► ►§4.29(ii) Complex Arguments
►The conformal mapping is obtainable from Figure 4.15.7 by rotating both the -plane and the -plane through an angle , compare (4.28.8). ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …17: 4.28 Definitions and Periodicity
§4.28 Definitions and Periodicity
… ►Relations to Trigonometric Functions
… ►As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions. ►Periodicity and Zeros
►The functions and have period , and has period . …18: 4.1 Special Notation
integers. | |
… |
19: 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.
Originally this equation appeared with in the second term, rather than .
Reported 2010-04-02.