modulus%20and%20phase

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1: 7.23 Tables

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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2: 10.75 Tables

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The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_0\left(x\right)$ to 15D, $J_1\left(x\right)$, $J_2\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$ to 10D, $Y_2\left(x\right)$ to 8D, $x=0(.1)17.5$; $Y_n\left(x\right)-(2/\pi )J_n\left(x\right)\ln x$, $n=0,1$, $x=0(.1)2$, 8D; $J_n\left(x\right)$, $Y_n\left(x\right)$, $n=3(1)9$, $x=0(.2)20$, 5D or 5S; $J_n\left(x\right)$, $Y_n\left(x\right)$, $n=0(1)20(10)50,100$, $x=1,2,5,10,50,100$, 10S; modulus and phase functions $\sqrt{x}{M}_n\left(x\right)$, ${\theta }_n\left(x\right)-x$, $n=0,1,2$, $1/x=0(.01)0.1$, 8D.

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3: 9.18 Tables

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Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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4: 22.3 Graphics

… ►In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. … ►

§22.3(iv) Complex $k$

… ►In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 22.3.24: $\mathrm{sn}(x+\mathrm{i}y,k)$ for $-4\le x\le 4$, $0\le y\le 8$, $k=1+\frac{1}{2}\mathrm{i}$. … Magnify 3D Help

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Figure 22.3.25: $\mathrm{sn}(5,k)$ as a function of complex $k^2$, $-1\le \mathrm{\Re }\left(k^2\right)\le 3.5$, $-1\le \mathrm{\Im }\left(k^2\right)\le 1$. … Magnify 3D Help

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5: 5.11 Asymptotic Expansions

… ►As $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$, … ►Wrench (1968) gives exact values of $g_k$ up to $g_{20}$. … ►If $z$ is complex, then the remainder terms are bounded in magnitude by ${\sec }^{2n}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.1), and ${\sec }^{2n+1}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.2), times the first neglected terms. … ►For this result and a similar bound for the sector $\frac{1}{2}\pi \le \mathrm{ph}z\le \pi$ see Boyd (1994). … ►If $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$, then …

6: 9.7 Asymptotic Expansions

… ►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►

9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

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See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.6 $${\mathrm{Ai}}^{\prime }\left(z\right)\sim -\frac{z^{1/4}{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

§9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E6

Encodings:

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See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.7 $$\mathrm{Bi}\left(z\right)\sim \frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.16), pp. 393 and 414)

Referenced by:

(9.10.5), (9.10.7), §9.7(iii), §9.7(iii), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E7

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Annotations for §9.7(ii), §9.7 and Ch.9

… ►where $\xi =\frac{2}{3}{x}^{3/2}$. …

7: 36.5 Stokes Sets

… ►The Stokes set consists of the rays $\mathrm{ph}x=\pm 2\pi /3$ in the complex $x$-plane. … ►

36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

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See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

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Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►For , there are two solutions $u$, provided that $|Y|>{(\frac{2}{5})}^{1/2}$. … ►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the $z$-axis by $2\pi /3$. …

8: 18.39 Applications in the Physical Sciences

… ►Then $\omega =2\pi \nu =\sqrt{k/m}$ is the circular frequency of oscillation (with $\nu$ the ordinary frequency), independent of the amplitude of the oscillations. …With the normalization factor ${\left(c{h}_n\right)}^{-1/2}$ the ${\psi }_n$ are orthonormal in $L^2(ℝ,dx)$. … ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials ${𝐋}_{n+l}^{2l+1}({\rho }_n)$. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. … ►For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials ($\alpha =\beta =0$) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974). …

9: 9.9 Zeros

… ►They lie in the sectors and , and are denoted by ${\beta }_k$, ${\beta }_k^{\prime }$, respectively, in the former sector, and by $\overline{{\beta }_k}$, $\overline{{\beta }_k^{\prime }}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\mathrm{\dots }.$ See §9.3(ii) for visualizations. … ►

§9.9(ii) Relation to Modulus and Phase

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9.9.1 $$\theta \left(a_k\right)=\varphi \left(a_{k+1}^{\prime }\right)=k\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $a_k^{\prime }$: $k$th zero of Airy ${\mathrm{Ai}}^{\prime }$ and $k$: nonnegative integer

Source:

Olver (1997b, (5.04), p. 404)

Referenced by:

(9.9.3), (9.9.4)

Permalink:

http://dlmf.nist.gov/9.9.E1

Encodings:

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Annotations for §9.9(ii), §9.9 and Ch.9

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Table 9.9.3: Complex zeros of $\mathrm{Bi}$.

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	${\mathrm{e}}^{-\pi \mathrm{i}/3}{\beta }_k$		${\mathrm{Bi}}^{\prime }\left({\beta }_k\right)$
$k$	modulus	phase	modulus	phase
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Table 9.9.4: Complex zeros of ${\mathrm{Bi}}^{\prime }$.

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	${\mathrm{e}}^{-\pi \mathrm{i}/3}{\beta }_k^{\prime }$		$\mathrm{Bi}\left({\beta }_k^{\prime }\right)$
$k$	modulus	phase	modulus	phase
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10: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

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Annotations for §11.6(i), §11.6 and Ch.11

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11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

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See also:

Annotations for §11.6(ii), §11.6 and Ch.11

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§11.6(iii) Large $|\nu |$, Fixed $z/\nu$

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$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

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11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

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See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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