modulus and phase

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21: 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

… ►

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

► ►►►

	${\mathrm{e}}^{-\pi \mathrm{i}/3}{\beta }_k$		${\mathrm{Bi}}^{\prime }\left({\beta }_k\right)$
$k$	modulus	phase	modulus	phase
…

►

►

Table 9.9.4: Complex zeros of ${\mathrm{Bi}}^{\prime }$.

► ►►►

	${\mathrm{e}}^{-\pi \mathrm{i}/3}{\beta }_k^{\prime }$		$\mathrm{Bi}\left({\beta }_k^{\prime }\right)$
$k$	modulus	phase	modulus	phase
…

►

22: 15.12 Asymptotic Approximations

… ►

$\mathrm{\Re }z=\frac{1}{2}$ and $\left|\mathrm{ph}c\right|\le \pi -\delta$.

… ►If , then (15.12.3) applies when $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$. … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$, … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$, … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$, …

23: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.3 $$I_{\nu }^{\prime }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.3

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

… ►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). … ►as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$. … ►as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$. … ►where $\chi (\mathrm{\ell })={\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+\frac{1}{2}\right)$; see §9.7(i). …

24: 9.12 Scorer Functions

… ►

9.12.9 $$\mathrm{Hi}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right),\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right),$$ $|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase and $z$: complex variable

Source:

Olver (1997b, p. 432)

Permalink:

http://dlmf.nist.gov/9.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(iv), §9.12 and Ch.9

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9.12.10 $${\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Hi}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right),\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{3}\pi$.

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase and $z$: complex variable

Source:

Olver (1997b, p. 432)

Permalink:

http://dlmf.nist.gov/9.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(iv), §9.12 and Ch.9

… ►

9.12.22 $$\mathrm{Hi}\left(-z\right)=\frac{4z^2}{3^{3/2}{\pi }^2}{\int }_0^{\mathrm{\infty }}\frac{K_{1/3}\left(t\right)}{{\zeta }^2+t^2}dt,$$ ,

ⓘ

Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\zeta$: variable

Source:

Gordon (1970, Appendix A)

Permalink:

http://dlmf.nist.gov/9.12.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

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9.12.29 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k}+\frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$.

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/9.12.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

…

25: 10.17 Asymptotic Expansions for Large Argument

… ►

10.17.7 $$z^{\frac{1}{2}}=\exp \left(\frac{1}{2}\ln |z|+\frac{1}{2}\mathrm{i}\mathrm{ph}z\right).$$

ⓘ

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

►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 $k\ge 2$, … ►

10.17.15

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable and $\chi (\mathrm{\ell })$

Referenced by:

§10.17(iv)

Permalink:

http://dlmf.nist.gov/10.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

►where $\chi (\mathrm{\ell })={\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}\mathrm{\ell }+\frac{1}{2}\right)$; see §9.7(i). …

26: 8.11 Asymptotic Approximations and Expansions

… ►

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

ⓘ

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

…

27: 32.11 Asymptotic Approximations for Real Variables

… ►

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

ⓘ

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

…

28: 7.14 Integrals

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7.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}at}\mathrm{erfc}\left(bt\right)dt=\frac{1}{a\sqrt{\pi }}F\left(\frac{a}{b}\right)+\frac{\mathrm{i}}{2a}\left(1-{\mathrm{e}}^{-{(a/b)}^2}\right),$$ $a\in ℂ$, .

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{ph}$: phase

A&S Ref:

7.4.18 (in different form)

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

… ►

7.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\left(bt\right)dt=\frac{1}{a}{\mathrm{e}}^{a^2/(4b^2)}\mathrm{erfc}\left(\frac{a}{2b}\right),$$ $\mathrm{\Re }a>0$, ,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.17

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

…

29: 9.2 Differential Equation

… ►

Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.

► ►►►►►

Pair	Interval or Region
…
$\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)$	$-\frac{1}{3}\pi \le \mathrm{ph}z\le \pi$
$\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)$	$-\pi \le \mathrm{ph}z\le \frac{1}{3}\pi$
$\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)$	$|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi$

►

…

30: 25.10 Zeros

… ►

25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

►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. … ►

25.10.3 $$Z(t)=2\sum _{n=1}^m\frac{\cos \left(\vartheta (t)-t\ln n\right)}{n^{1/2}}+R(t),$$ $m=\lfloor \sqrt{t/(2\pi )}\rfloor$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\lfloor x\rfloor$: floor of $x$ , $\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $n$: nonnegative integer, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

Riemann–Siegel formula

Source:

Titchmarsh (1986b, (4.17.4), p. 89)

Referenced by:

§25.18(i), Erratum (V1.1.4) for Equation (25.10.3)

Permalink:

http://dlmf.nist.gov/25.10.E3

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint $m=\lfloor \sqrt{t/(2\pi )}\rfloor$ was added.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.10(ii), §25.10 and Ch.25

►where $R(t)=O\left(t^{-1/4}\right)$ as $t\to \mathrm{\infty }$. … ►

25.10.4 $$R(t)={(-1)}^{m-1}{\left(\frac{2\pi }{t}\right)}^{1/4}\frac{\cos \left(t-(2m+1)\sqrt{2\pi t}-\frac{1}{8}\pi \right)}{\cos \left(\sqrt{2\pi t}\right)}+O\left(t^{-3/4}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $m$: nonnegative integer

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.17.5), p. 89)

Referenced by:

§25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii)

Permalink:

http://dlmf.nist.gov/25.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(ii), §25.10 and Ch.25

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