modulus and phase

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1: 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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2: 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 $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,3\right)$. …

3: 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). …

4: 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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5: About Color Map

… ►In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of $\pi /2$ correponding to the real and imaginary axes, at more immediately recognizable colors. … ►We therefore use a piecewise linear mapping as illustrated below, that takes phase $0$ to red, $\pi /2$ to yellow, $\pi$ to cyan and $3\pi /2$ to blue. …

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: 33.11 Asymptotic Expansions for Large $\rho$

… ►where ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$ is defined by (33.2.9), and $a$ and $b$ are defined by (33.8.3). … ►

$F_{\mathrm{\ell }}(\eta ,\rho )=g(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+f(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}(\eta ,\rho )=f(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}-g(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=\widehat{g}(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+\widehat{f}(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

… ►Here $f_0=1$, $g_0=0$, ${\widehat{f}}_0=0$, ${\widehat{g}}_0=1-(\eta /\rho )$, and for $k=0,1,2,\mathrm{\dots }$, …

8: 10.27 Connection Formulas

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10.27.6 $$I_{\nu }\left(z\right)={\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{J}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.3 (modified)

Referenced by:

§10.26(ii), §10.27, §10.27, §10.31, §10.33, §10.35, §10.39, §10.41(i), §10.42, §10.43(ii), §10.43(iv), §10.44(i), §10.44(ii), §10.44(iii), §10.47(iv), §10.67(i), §11.4(i)

Permalink:

http://dlmf.nist.gov/10.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.7 $$I_{\nu }\left(z\right)=\frac{1}{2}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}\left(H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)+H_{\nu }^{(2)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$.

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.9 $$\pi \mathrm{i}{J}_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.67(i)

Permalink:

http://dlmf.nist.gov/10.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.10 $$-\pi Y_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.11 $$Y_{\nu }\left(z\right)={\mathrm{e}}^{\pm (\nu +1)\pi \mathrm{i}/2}{I}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right)-(2/\pi ){\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right),$$ $-\frac{1}{2}\pi \le \pm \mathrm{ph}z\le \pi$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.5 (modified)

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/10.27.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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9: 9.3 Graphics

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§9.3(i) Real Variable

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10: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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$$F_{\mathrm{\ell }}(\eta ,\rho )=\sin \left({\theta }_{\mathrm{\ell }}(\eta ,\rho )\right)+o\left(1\right),$$

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$$G_{\mathrm{\ell }}(\eta ,\rho )=\cos \left({\theta }_{\mathrm{\ell }}(\eta ,\rho )\right)+o\left(1\right),$$

… ►where ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$ is defined by (33.2.9). … ►

${\sigma }_0\left(\eta \right)=\eta (\ln \eta -1)+\frac{1}{4}\pi +o\left(1\right),$

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${\sigma }_0\left(\eta \right)=\eta (\ln \left(-\eta \right)-1)-\frac{1}{4}\pi +o\left(1\right),$

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