modulus and phase functions

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11: 4.2 Definitions

… ►For $\mathrm{ph}z$ see §1.9(i). … ►

§4.2(iii) The Exponential Function

… ►The general value of the phase is given by … ►

§4.2(iv) Powers

… ► …

12: 11.6 Asymptotic Expansions

… ►

§11.6(i) Large $|z|$, Fixed $\nu$

… ►

§11.6(ii) Large $|\nu |$, Fixed $z$

… ► ►

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

… ►Here …

13: 9.9 Zeros

… ►

§9.9(ii) Relation to Modulus and Phase

…

14: 4.9 Continued Fractions

… ►

§4.9(i) Logarithms

►

4.9.1 $$\ln \left(1+z\right)=\frac{z}{1+}\frac{z}{2+}\frac{z}{3+}\frac{4z}{4+}\frac{4z}{5+}\frac{9z}{6+}\frac{9z}{7+}\mathrm{\cdots },$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.39

Permalink:

http://dlmf.nist.gov/4.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

… ►

§4.9(ii) Exponentials

… ►

$$=1+\frac{z}{1-(z/2)+}\frac{z^2/(4\cdot 3)}{1+}\frac{z^2/(4\cdot 15)}{1+}\frac{z^2/(4\cdot 35)}{1+}\mathrm{\cdots }\frac{z^2/(4(4n^2-1))}{1+}\mathrm{\cdots }$$

… ►For other continued fractions involving the exponential function see Lorentzen and Waadeland (1992, pp. 563–564). …

15: 8.11 Asymptotic Approximations and Expansions

… ►

8.11.5 $$P(a,z)\sim \frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(1+a\right)}\sim {(2\pi a)}^{-\frac{1}{2}}{\mathrm{e}}^{a-z}{(z/a)}^a,$$ $a\to \mathrm{\infty }$, $|\mathrm{ph}a|\le \pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter and $\delta$: small positive constant

Referenced by:

§8.11(ii)

Permalink:

http://dlmf.nist.gov/8.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(ii), §8.11 and Ch.8

…

16: 9.7 Asymptotic Expansions

… ►

9.7.9 $$\mathrm{Ai}\left(-z\right)\sim \frac{1}{\sqrt{\pi }{z}^{1/4}}\left(\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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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, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.08), pp. 392 and 413)

Referenced by:

§9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E9

Encodings:

pMML, png, TeX

See also:

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

►

9.7.10 $${\mathrm{Ai}}^{\prime }\left(-z\right)\sim \frac{z^{1/4}}{\sqrt{\pi }}\left(\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}-\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

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

Source:

Olver (1997b, (1.09), pp. 392 and 413)

Permalink:

http://dlmf.nist.gov/9.7.E10

Encodings:

pMML, png, TeX

See also:

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

►

9.7.11 $$\mathrm{Bi}\left(-z\right)\sim \frac{1}{\sqrt{\pi }{z}^{1/4}}\left(-\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k}}{{\zeta }^{2k}}+\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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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, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.17), pp. 393 and 414)

Permalink:

http://dlmf.nist.gov/9.7.E11

Encodings:

pMML, png, TeX

See also:

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

►

9.7.12 $${\mathrm{Bi}}^{\prime }\left(-z\right)\sim \frac{z^{1/4}}{\sqrt{\pi }}\left(\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$.

ⓘ

Symbols:

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

Source:

Olver (1997b, (1.18), pp. 393 and 414)

Referenced by:

§9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E12

Encodings:

pMML, png, TeX

See also:

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

►

9.7.13 $$\mathrm{Bi}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/3}\right)\sim \sqrt{\frac{2}{\pi }}\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}/6}}{z^{1/4}}\left(\cos \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \mp \frac{1}{2}\mathrm{i}\ln 2\right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

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

Proof sketch:

See Olver (1997b, pp. 392–393 and 413–414).

Permalink:

http://dlmf.nist.gov/9.7.E13

Encodings:

pMML, png, TeX

See also:

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

…

17: 10.17 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for other ranges of $\mathrm{ph}z$ can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4). … ►Also, $b_0(\nu )=1$, $b_1(\nu )=(4{\nu }^2+3)/8$, and for $k\ge 2$, … ►

§10.17(iii) Error Bounds for Real Argument and Order

… ►

§10.17(v) Exponentially-Improved Expansions

…

18: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

… ►

§33.10(i) Large $\rho$

… ►

33.10.2 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim \exp \left(\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )\right),$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\sim$: asymptotic equality, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.6.5

Permalink:

http://dlmf.nist.gov/33.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.10(i), §33.10 and Ch.33

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

§33.10(ii) Large Positive $\eta$

… ►

§33.10(iii) Large Negative $\eta$

…

19: 25.10 Zeros

… ►

§25.10(i) Distribution

… ► … ►where …is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. … ►where $R(t)=O\left(t^{-1/4}\right)$ as $t\to \mathrm{\infty }$. …

20: 12.14 The Function $W(a,x)$

… ►

§12.14(x) Modulus and Phase Functions

… ►

12.14.37 $$k^{-1/2}W(a,x)+\mathrm{i}{k}^{1/2}W(a,-x)=\tilde{F}(a,x){\mathrm{e}}^{\mathrm{i}\tilde{\theta }(a,x)},$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/12.14.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(x), §12.14 and Ch.12

►

12.14.38 $$k^{-1/2}{W}^{\prime }(a,x)+\mathrm{i}{k}^{1/2}{W}^{\prime }(a,-x)=-\tilde{G}(a,x){\mathrm{e}}^{\mathrm{i}\tilde{\psi }(a,x)},$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/12.14.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(x), §12.14 and Ch.12

… ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …