Bose–Einstein phase transition

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11: Bibliography B

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Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.

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External Links:

ISSN 0305-4470, Document, MathReview (J. S. Joel), ZentralBlatt

Referenced by:

§36.9.

Encodings:

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Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

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External Links:

Document

Referenced by:

§29.19(i).

Encodings:

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The determination of phases and amplitudes of wave functions. Proc. Phys. Soc. 81 (3), pp. 442–452.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

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12: Software Index

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	Open Source													With Book						Commercial
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25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
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13: Bibliography D

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The Bose-Einstein integrals ${ℬ}_p(\eta )={(p!)}^{-1}{\int }_0^{\mathrm{\infty }}{ϵ}^p{(e^{ϵ-\eta }-1)}^{-1}𝑑ϵ$ . Appl. Sci. Res. B. 6, pp. 240–244.

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External Links:

ISSN 0003-6994, MathReview Entry, ZentralBlatt

Referenced by:

(25.12.15), §25.12(iii), §25.12.

Encodings:

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14: Bibliography G

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On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

Encodings:

BibTeX

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15: 27.17 Other Applications

… ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

16: 10.19 Asymptotic Expansions for Large Order

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§10.19(iii) Transition Region

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$$J_{\nu }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\sim \frac{2^{\frac{1}{3}}}{{\nu }^{\frac{1}{3}}}\mathrm{Ai}\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{2}{3}}}{\nu }{\mathrm{Ai}}^{\prime }\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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$$Y_{\nu }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\sim -\frac{2^{\frac{1}{3}}}{{\nu }^{\frac{1}{3}}}\mathrm{Bi}\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}-\frac{2^{\frac{2}{3}}}{\nu }{\mathrm{Bi}}^{\prime }\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$.

… ►with sectors of validity $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}\nu \le \frac{3}{2}\pi -\delta$. … ►with sectors of validity $-\frac{1}{2}\pi +\delta \le \mathrm{ph}\nu \le \frac{3}{2}\pi -\delta$ and $-\frac{3}{2}\pi +\delta \le \mathrm{ph}\nu \le \frac{1}{2}\pi -\delta$, respectively. …

17: 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. …

18: 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 $4\le \eta \le \mathrm{\infty }$. …

19: 36.7 Zeros

… ►The zeros in Table 36.7.1 are points in the $𝐱=(x,y)$ plane, where $\mathrm{ph}{\mathrm{\Psi }}_2\left(𝐱\right)$ is undetermined. … ►The zeros are lines in $𝐱=(x,y,z)$ space where $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$ is undetermined. …, $y=0$), the number of rings in the $m$th row, measured from the origin and before the transition to hairpins, is given by …

20: 1.6 Vectors and Vector-Valued Functions

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Einstein Summation Convention

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