relation to modulus and phase

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11: 32.11 Asymptotic Approximations for Real Variables

… ►where $\mathrm{\Gamma }$ is the gamma function (§5.2(i)), and the branch of the $\mathrm{ph}$ function is immaterial. … ►where $\lambda$ is an arbitrary constant such that , and …The connection formulas relating (32.11.25) and (32.11.26) are … ►Now suppose $x\to -\mathrm{\infty }$. …and the branch of the $\mathrm{ph}$ function is immaterial. …

12: 33.23 Methods of Computation

§33.23 Methods of Computation

… ►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►

§33.23(iv) Recurrence Relations

►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\mathrm{\ell }$, provided that the recurrence is carried out in a stable direction (§3.6). … ► (12) $(\rho -c)/c$ should be $(\rho -c)/\rho$). …

13: 25.10 Zeros

… ►Calculations relating to the zeros on the critical line make use of the real-valued function … ►

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

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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:

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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. … ►Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp \left(\mathrm{i}\vartheta (t)\right)$ to obtain the Riemann–Siegel formula: …where $R(t)=O\left(t^{-1/4}\right)$ as $t\to \mathrm{\infty }$. …

14: 9.13 Generalized Airy Functions

… ►Swanson and Headley (1967) define independent solutions $A_n\left(z\right)$ and $B_n\left(z\right)$ of (9.13.1) by … ►Their relations to the functions $A_n\left(z\right)$ and $B_n\left(z\right)$ are given by … ►The function on the right-hand side is recessive in the sector $-(2j-1)\pi /m\le \mathrm{ph}z\le (2j+1)\pi /m$, and is therefore an essential member of any numerically satisfactory pair of solutions in this region. … ►When $\alpha$ is a positive integer the relation of these functions to $W_m(t)$, $W_m(-t)$ is as follows: … ►When $p$ is not an integer the branch of $t^{-p}$ in (9.13.25) is usually chosen to be $\exp \left(-p(\ln |t|+\mathrm{i}\mathrm{ph}t)\right)$ with . …

15: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

►

§15.9(i) Orthogonal Polynomials

… ►

Jacobi

… ►

Legendre

… ►

Meixner

…

16: 36.7 Zeros

… ►The zeros in Table 36.7.1 are points in the $𝐱=(x,y)$ plane, where $\mathrm{ph}{\mathrm{\Psi }}_2\left(𝐱\right)$ is undetermined. … ►The zeros are lines in $𝐱=(x,y,z)$ space where $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$ is undetermined. …The rings are almost circular (radii close to $(\Delta x)/9$ and varying by less than 1%), and almost flat (deviating from the planes $z_n$ by at most $(\Delta z)/36$). …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by …

17: 8.19 Generalized Exponential Integral

… ►

§8.19(i) Definition and Integral Representations

… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►with $|\mathrm{ph}z|\le \pi$ in both equations. … ►

§8.19(v) Recurrence Relation and Derivatives

… ►

§8.19(vi) Relation to Confluent Hypergeometric Function

…

18: 9.11 Products

… ►For example, $w={\mathrm{Ai}}^2\left(z\right)$, $\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right)$, $\mathrm{Ai}\left(z\right)\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)$, $M^2\left(z\right)$. … ►

9.11.4 $${\mathrm{Ai}}^2\left(z\right)+{\mathrm{Bi}}^2\left(z\right)=\frac{1}{{\pi }^{3/2}}{\int }_0^{\mathrm{\infty }}\exp \left(zt-\frac{1}{12}{t}^3\right)t^{-1/2}dt.$$

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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 $x$ , $\exp z$: exponential function, $\int$: integral and $z$: complex variable

Source:

Muldoon (1977, p. 32, extended to complex $z$ by analytic continuation)

Referenced by:

§9.5(ii)

Permalink:

http://dlmf.nist.gov/9.11.E4

Encodings:

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See also:

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

… ►For related integrals see Gordon (1969, Appendix B). … ►

9.11.15 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}{\mathrm{Ai}}^2\left(t\right)dt=\frac{2\mathrm{\Gamma }\left(\alpha \right)}{{\pi }^{1/2}{12}^{(2\alpha +5)/6}\mathrm{\Gamma }\left(\frac{1}{3}\alpha +\frac{5}{6}\right)},$$ $\mathrm{\Re }\alpha >0$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Source:

Laurenzi (1993, p. 904)

Permalink:

http://dlmf.nist.gov/9.11.E15

Encodings:

pMML, png, TeX

See also:

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

… ►

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:

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See also:

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

…

19: 25.12 Polylogarithms

… ►When $z={\mathrm{e}}^{\mathrm{i}\theta }$, $0\le \theta \le 2\pi$, (25.12.1) becomes … ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►Further properties include …and … ►In terms of polylogarithms …

20: 10.9 Integral Representations

… ►

Poisson’s and Related Integrals

… ►

Schläfli’s and Related Integrals

… ►

Mehler–Sonine and Related Integrals

… ►When , … ►See Paris and Kaminski (2001, p. 116) for related results. …