# pictures of phase

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## 3 matching pages

##### 1: 36.3 Visualizations of Canonical Integrals

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##### 2: Errata

The Olver hypergeometric function $\mathbf{F}(a,b;c;z)$, 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.

In §10.37, it was originally stated incorrectly that (10.37.1) holds for $$. The claim has been updated to $|\mathrm{ph}z|\le \frac{1}{2}\pi $.

*Reported 2017-11-14 by Gergő Nemes.*

Scales were corrected in all figures. The interval $-8.4\le \frac{x-y}{\sqrt{2}}\le 8.4$ was replaced by $-12.0\le \frac{x-y}{\sqrt{2}}\le 12.0$ and $-12.7\le \frac{x+y}{\sqrt{2}}\le 4.2$ replaced by $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0$. 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 $|{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,0\right)|$.

(a) Density plot. | (b) 3D plot. |

Figure 36.3.10: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,1\right)|$.

(a) Density plot. | (b) 3D plot. |

Figure 36.3.11: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,2\right)|$.

(a) Density plot. | (b) 3D plot. |

Figure 36.3.12: Modulus of hyperbolic umbilic canonical integral function $|{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,3\right)|$.

*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 $-8.4\le \frac{x-y}{\sqrt{2}}\le 8.4$ was replaced by $-12.0\le \frac{x-y}{\sqrt{2}}\le 12.0$ and $-12.7\le \frac{x+y}{\sqrt{2}}\le 4.2$ replaced by $-18.0\le \frac{x+y}{\sqrt{2}}\le 6.0$. 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 $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,0\right)$.

(a) Contour plot. | (b) Density plot. |

Figure 36.3.19: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,1\right)$.

(a) Contour plot. | (b) Density plot. |

Figure 36.3.20: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,2\right)$.

(a) Contour plot. | (b) Density plot. |

Figure 36.3.21: Phase of hyperbolic umbilic canonical integral $\mathrm{ph}{\mathrm{\Psi}}^{(\mathrm{H})}\left(x,y,3\right)$.

*Reported 2016-09-28.*

Originally the constraint, $$, was incorrectly written as, $$.

*Reported 2015-05-20 by Richard Paris.*