modulus and phase functions

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1: 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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§9.8(iv) Asymptotic Expansions

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2: 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

… ►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …

3: 10.18 Modulus and Phase Functions

§10.18 Modulus and Phase Functions

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§10.18(i) Definitions

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§10.18(ii) Basic Properties

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§10.18(iii) Asymptotic Expansions for Large Argument

… ►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

4: 12.19 Tables

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Miller (1955) includes $W(a,x)$, $W(a,-x)$, and reduced derivatives for $a=-10(1)10$, $x=0(.1)10$, 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

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Fox (1960) includes modulus and phase functions for $W(a,x)$ and $W(a,-x)$, and several auxiliary functions for $x^{-1}=0(.005)0.1$, $a=-10(1)10$, 8S.

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5: 10.3 Graphics

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§10.3(i) Real Order and Variable

►For the modulus and phase functions $M_{\nu }\left(x\right)$, ${\theta }_{\nu }\left(x\right)$, $N_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ see §10.18. … ►

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Figure 10.3.4: ${\theta }_5\left(x\right)$, ${\varphi }_5\left(x\right)$, $0\le x\le 15$. Magnify

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6: 33.13 Complex Variable and Parameters

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33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

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7: 36.3 Visualizations of Canonical Integrals

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§36.3(i) Canonical Integrals: Modulus

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

8: 32.11 Asymptotic Approximations for Real Variables

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32.11.24 $$s={\left(\exp \left(\pi d^2\right)-1\right)}^{1/2}\exp \left(\mathrm{i}\left(\frac{3}{2}{d}^2\ln 2-\frac{1}{4}\pi +\chi -\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{i}{d}^2\right)\right)\right).$$

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Defines:

$s$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\chi$: constant and $d$: constant

Permalink:

http://dlmf.nist.gov/32.11.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iii), §32.11 and Ch.32

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9: 9.11 Products

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9.11.19 $${\int }_0^{\mathrm{\infty }}\frac{dt}{{\mathrm{Ai}}^2\left(t\right)+{\mathrm{Bi}}^2\left(t\right)}={\int }_0^{\mathrm{\infty }}\frac{tdt}{\mathrm{Ai}_{}^{\prime }{}_{}{}^2\left(t\right)+\mathrm{Bi}_{}^{\prime }{}_{}{}^2\left(t\right)}=\frac{{\pi }^2}{6}.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Extend the definitions of §9.8(i) to positive values of $x$, obtain the indefinite integrals of $1/M^2\left(x\right)$ and $x/N^2\left(x\right)$ via the first two of (9.8.14), then combine the values of $\theta \left(0\right)$ and $\varphi \left(0\right)$ given in §9.8(i) with $\theta \left(+\mathrm{\infty }\right)=\varphi \left(+\mathrm{\infty }\right)=0$ obtained from (9.8.4), (9.8.8), and §9.7(ii). (Communicated by M.E. Muldoon.)

Permalink:

http://dlmf.nist.gov/9.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(v), §9.11 and Ch.9

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10: 12.2 Differential Equations

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§12.2(vi) Solution $\overline{U}(a,x)$; Modulus and Phase Functions

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12.2.22 $$U(a,x)+\mathrm{i}\overline{U}(a,x)=F(a,x){\mathrm{e}}^{\mathrm{i}\theta (a,x)},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $F(a,x)$: modulus and $\theta (a,x)$: phase

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.2.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(vi), §12.2 and Ch.12

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12.2.23 $$U^{\prime }(a,x)+\mathrm{i}{\overline{U}}^{\prime }(a,x)=-G(a,x){\mathrm{e}}^{\mathrm{i}\psi (a,x)},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $x$: real variable, $a$: real or complex parameter, $G(a,x)$: modulus and $\psi (a,x)$: phase

Permalink:

http://dlmf.nist.gov/12.2.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(vi), §12.2 and Ch.12

… ►For properties of the modulus and phase functions, including differential equations, see Miller (1955, pp. 72–73). …