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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 ${\sigma }_0\left(\eta \right)=\mathrm{ph}\mathrm{\Gamma }\left(1+\mathrm{i}\eta \right)$ (see (33.2.10)) for the ranges $0\le \eta \le 2$, $2\le \eta \le 4$, and $4\le \eta \le \mathrm{\infty }$. …

3: 36.3 Visualizations of Canonical Integrals

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§36.3(ii) Canonical Integrals: Phase

►In Figure 36.3.13(a) points of confluence of phase contours are zeros of ${\mathrm{\Psi }}_2\left(x,y\right)$; similarly for other contour plots in this subsection. … ►

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Figure 36.3.13: Phase of Pearcey integral $\mathrm{ph}{\mathrm{\Psi }}_2\left(x,y\right)$.

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Figure 36.3.14: Density plots of phase of swallowtail canonical integrals.

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

4: 9.8 Modulus and Phase

§9.8 Modulus and Phase

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§9.8(i) Definitions

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§9.8(ii) Identities

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§9.8(iii) Monotonicity

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5: About Color Map

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Phase Mappings

►By painting the surfaces with a color that encodes the phase, $\mathrm{ph}f$, both the magnitude and phase of complex valued functions can be displayed. We offer two options for encoding the phase. ►

Four Color Phase Mapping

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Continuous Phase Mapping

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6: Sidebar 5.SB1: Gamma & Digamma Phase Plots

Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►The color encoded phases of $\mathrm{\Gamma }\left(z\right)$ (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 $\mathrm{\Gamma }\left(z\right)$ 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 Modulus and Phase Functions

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§10.68(i) Definitions

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§10.68(ii) Basic Properties

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§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

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6.9.1 $$E_1\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z+}\frac{1}{1+}\frac{1}{z+}\frac{2}{1+}\frac{2}{z+}\frac{3}{1+}\frac{3}{z+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

5.1.22 (special case of)

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.9 and Ch.6

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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 Modulus and Phase Functions

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§10.18(i) Definitions

… ►where $M_{\nu }\left(x\right)$ $(>0)$, $N_{\nu }\left(x\right)$ $(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $\nu$ and $x$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ fixed by … ►

§10.18(ii) Basic Properties

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§10.18(iii) Asymptotic Expansions for Large Argument

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