# phase

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## 1—10 of 187 matching pages

##### 1: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
###### Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …
##### 2: 33.25 Approximations
###### §33.25 Approximations
Cody and Hillstrom (1970) provides rational approximations of the phase shift ${\sigma_{0}}\left(\eta\right)=\operatorname{ph}\Gamma\left(1+\mathrm{i}\eta\right)$ (see (33.2.10)) for the ranges $0\leq\eta\leq 2$, $2\leq\eta\leq 4$, and $4\leq\eta\leq\infty$. …
##### 3: 36.3 Visualizations of Canonical Integrals
###### §36.3(ii) Canonical Integrals: Phase
In Figure 36.3.13(a) points of confluence of phase contours are zeros of $\Psi_{2}\left(x,y\right)$; similarly for other contour plots in this subsection. …
##### 5: About Color Map
###### Phase Mappings
By painting the surfaces with a color that encodes the phase, $\operatorname{ph}f$, both the magnitude and phase of complex valued functions can be displayed. We offer two options for encoding the phase.
##### 6: Sidebar 5.SB1: Gamma & Digamma Phase Plots
###### Sidebar 5.SB1: Gamma & Digamma Phase Plots
The color encoded phases of $\Gamma\left(z\right)$ (above) and $\psi\left(z\right)$ (below), are constrasted in the negative half of the complex plane. In the upper half of the image, the poles of $\Gamma\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. … Phase changes around the zeros are of opposite sign to those around the poles. …
##### 7: 10.68 Modulus and Phase Functions
###### §10.68(iii) Asymptotic Expansions for Large Argument
However, care needs to be exercised with the branches of the phases. …
##### 8: 6.9 Continued Fraction
6.9.1 $E_{1}\left(z\right)=\cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+% \cfrac{2}{z+\cfrac{3}{1+\cfrac{3}{z+}}}}}}}\cdots,$ $|\operatorname{ph}z|<\pi$.
##### 9: 25.17 Physical Applications
See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …
##### 10: 10.18 Modulus and Phase Functions
###### §10.18(i) Definitions
where $M_{\nu}\left(x\right)$ $(>0)$, $N_{\nu}\left(x\right)$ $(>0)$, $\theta_{\nu}\left(x\right)$, and $\phi_{\nu}\left(x\right)$ are continuous real functions of $\nu$ and $x$, with the branches of $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ fixed by …