change of modulus

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21—30 of 88 matching pages

21: 23.6 Relations to Other Functions

… ►With

z=\ifrac{\pi u}{(2\omega_{1})}

, …

22: 8.27 Approximations

… ►

•

DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\mathrm{d}t$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$ . This takes the form $F(p,x)=4x/h(p,x)$ , approximately, where $h(p,x)=3(x^{2}-p)+\sqrt{(x^{2}-p)^{2}+8(x^{2}+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to\infty$ .

…

23: 9.5 Integral Representations

… ►

9.5.7

\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp\left(-z^% {\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}t^{3}\right)\mathrm{d}t,

|\operatorname{ph}z|<\pi

ⓘ

Symbols:: $\mathrm{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\zeta$ : change of variable
Source:: Copson (1963, (2.1))
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.5(ii), §9.5 and Ch.9

►

9.5.8

\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{\pi}(48% )^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t^{-% \ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\mathrm{d}t,

|\operatorname{ph}z|<\frac{2}{3}\pi

ⓘ

Symbols:: $\mathrm{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\zeta$ : change of variable
Source:: Follows from (9.6.21) and (13.4.4).
Referenced by:: §9.17(iii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §9.5(ii), §9.5 and Ch.9

…

24: 27.2 Functions

… ►Gauss and Legendre conjectured that

\pi\left(x\right)

is asymptotic to

x/\ln x

x\to\infty

: ►

27.2.3

\pi\left(x\right)\sim\frac{x}{\ln x}.

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\ln\NVar{z}$ : principal branch of logarithm function, $\pi\left(\NVar{x}\right)$ : number of primes not exceeding $x$ and $x$ : real number
Referenced by:: §25.16(i), §27.11
Permalink:: http://dlmf.nist.gov/27.2.E3
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.4

p_{n}\sim n\ln n.

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.2.E4
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.14

\Lambda\left(n\right)=\ln p,

n=p^{a}

ⓘ

Defines:: $\Lambda\left(\NVar{n}\right)$ : Mangoldt’s function
Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer, $p,p_{1},\ldots$ : prime numbers and $a$ : primitive root
Permalink:: http://dlmf.nist.gov/27.2.E14
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.2(i), §27.2 and Ch.27

…

25: 19.25 Relations to Other Functions

… ►then the five nontrivial permutations of

x, y, z

that leave

R_{F}

invariant change

k^{2}

(

=(z-y)/(z-x)

) into

1/k^{2}

{k^{\prime}}^{2}

1/{k^{\prime}}^{2}

-k^{2}/{k^{\prime}}^{2}

-{k^{\prime}}^{2}/k^{2}

, and

\sin\phi

(

=\sqrt{(z-x)/z}

) into

k\sin\phi

-i\tan\phi

-ik^{\prime}\tan\phi

(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin^{2}}\phi}

-ik\sin\phi/\sqrt{1-k^{2}{\sin^{2}}\phi}

. …

26: Errata

… ►

Other Changes

•

In Paragraph Steed’s Algorithm in §3.10(iii), a sentence was added to inform the reader of alternatives to Steed’s algorithm, namely the Lentz algorithm (see e.g., Lentz (1976)) and the modified Lentz algorithm (see e.g., Thompson and Barnett (1986)).
•

In Subsection 19.11(i), a sentence and unnumbered equation

$R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right),$

were added which indicate that care must be taken with the multivalued functions in (19.11.5). See (Cayley, 1961, pp. 103-106). This was suggested by Albert Groenenboom.
•

In Subsection 33.14(iv), there were several modifications. Just below (33.14.9), the constraint described in the text “ $\ell<(-\epsilon)^{-1/2}$ when $\epsilon<0$ ,” was removed. In Equation (33.14.13), the constraint $\epsilon_{1},\epsilon_{2}>0$ was added. In the line immediately below (33.14.13), it was clarified that $s\left(\epsilon,\ell;r\right)$ is $\exp\left(-r/n\right)$ times a polynomial in $r/n$ , instead of simply a polynomial in $r$ . In Equation (33.14.14), a second equality was added which relates $\phi_{n,\ell}(r)$ to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions $\phi_{n,\ell}$ , $n=\ell,\ell+1,\ldots$ , do not form a complete orthonormal system.

… ►

Other Changes

•

The factor on the right-hand side of Equation (10.9.26) containing $\cos(\mu-\nu)\theta$ has been been replaced with $\cos\left((\mu-\nu)\theta\right)$ to clarify the meaning.
•

In Paragraph Confluent Hypergeometric Functions in §10.16, several Whittaker confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.
•

In Equation (15.6.9), it was clarified that $\lambda\in\mathbb{C}$ .
•

Originally Equation (19.16.9) had the constraint $a,a^{\prime}>0$ . This constraint was replaced with $b_{1}+\cdots+b_{n}>a>0$ , $b_{j}\in\mathbb{R}$ . It therefore follows from Equation (19.16.10) that $a^{\prime}>0$ . The last sentence of Subsection 19.16(ii) was elaborated to mention that generalizations may also be found in Carlson (1977b). These were suggested by Bastien Roucariès.
•

In Section 19.25(vi), the Weierstrass lattice roots $e_{j},$ were labeled inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_{j}$ , and lattice invariants $g_{2}$ , $g_{3}$ , now link to their respective definitions (see §§23.2(i), 23.3(i)). This was reported by Felix Ospald.
•

In Equation (19.25.37), the Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.
•

In Equation (27.12.5), the term originally written as $\sqrt{\ln x}$ was rewritten as $(\ln x)^{1/2}$ to be consistent with other equations in the same subsection.

… ►

Other Changes

•

In §5.2(iii), three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection. This was suggested by Tom Koornwinder.
•

Originally named as a complementary error function, (7.2.3) has been renamed as the Faddeeva (or Faddeyeva) function $w\left(z\right)$ .
•

In §7.8, an inequality (7.8.8) was added at the end of this section. This is Pólya (1949, (1.5)) and was suggested by Roberto Iacono.
•

Originally the function $\chi$ , used in (9.7.3) and (9.7.4), was presented with argument given by a positive integer $n$ . It has now been clarified to be valid for argument given by a positive real number $x$ .
•

Bounds have been sharpened in §9.7(iii). The second paragraph now reads, “The $n$ th error term is bounded in magnitude by the first neglected term multiplied by $\chi(n+\sigma)+1$ where $\sigma=\frac{1}{6}$ for (9.7.7) and $\sigma=0$ for (9.7.8), provided that $n\geq 0$ in the first case and $n\geq 1$ in the second case.” Previously it read, “In (9.7.7) and (9.7.8) the $n$ th error term is bounded in magnitude by the first neglected term multiplied by $2\chi(n)\exp\left(\sigma\pi/(72\zeta)\right)$ where $\sigma=5$ for (9.7.7) and $\sigma=7$ for (9.7.8), provided that $n\geq 1$ in both cases.” In Equation (9.7.16)

9.7.16
$\mathrm{Bi}\left(x\right)\leq\frac{{\mathrm{e}^{\xi}}}{\sqrt{\pi}x^{1/4}}\left% (1+\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}\right),$

$\mathrm{Bi}'\left(x\right)\leq\frac{x^{1/4}{\mathrm{e}^{\xi}}}{\sqrt{\pi}}% \left(1+\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}\right),$

the bounds on the right-hand sides have been sharpened. The factors $\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}$ , $\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}$ , were originally given by $\frac{5\pi}{72\xi}\exp\left(\frac{5\pi}{72\xi}\right)$ , $\frac{7\pi}{72\xi}\exp\left(\frac{7\pi}{72\xi}\right)$ , respectively.
•

Bounds have been sharpened in §9.7(iv). The first paragraph now reads, “The $n$ th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17
$\begin{cases}1,&|\operatorname{ph}z|\leq\tfrac{1}{3}\pi,\\ \min\left(|\!\csc\left(\operatorname{ph}\zeta\right)|,\chi(n\!+\!\sigma)\!+\!1% \right),&\tfrac{1}{3}\pi\leq|\operatorname{ph}z|\leq\tfrac{2}{3}\pi,\\ \frac{\sqrt{2\pi(n\!+\!\sigma)}}{|\!\cos\left(\operatorname{ph}\zeta\right)|^{% n\!+\!\sigma}}+\chi(n\!+\!\sigma)\!+\!1,&\tfrac{2}{3}\pi\leq|\operatorname{ph}% z|<\pi,\end{cases}$

provided that $n\geq 0$ , $\sigma=\tfrac{1}{6}$ for (9.7.5) and $n\geq 1$ , $\sigma=0$ for (9.7.6).” Previously it read, “When $n\geq 1$ the $n$ th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17
$\begin{cases}2\exp\left(\dfrac{\sigma}{36|\zeta|}\right)&|\operatorname{ph}z|% \leq\tfrac{1}{3}\pi,\\ 2\chi(n)\exp\left(\dfrac{\sigma\pi}{72|\zeta|}\right)&\tfrac{1}{3}\pi\leq|% \operatorname{ph}z|\leq\tfrac{2}{3}\pi,\\ \dfrac{4\chi(n)}{|\cos\left(\operatorname{ph}\zeta\right)|^{n}}\exp\left(% \dfrac{\sigma\pi}{36|\Re\zeta|}\right)&\tfrac{2}{3}\pi\leq|\operatorname{ph}z|% <\pi.\end{cases}$

Here $\sigma=5$ for (9.7.5) and $\sigma=7$ for (9.7.6).”
•

In §10.8, a sentence was added just below (10.8.3) indicating that it is a rewriting of (16.12.1). This was suggested by Tom Koornwinder.
•

Equations (10.15.1), (10.38.1), have been generalized to include the additional cases of $\ifrac{\partial J_{-\nu}\left(z\right)}{\partial\nu}$ , $\ifrac{\partial I_{-\nu}\left(z\right)}{\partial\nu}$ , respectively.
•

The Kronecker delta symbols in Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9), have been moved furthest to the right, as is common convention for orthogonality relations.
•

The titles of §§14.5(ii), 14.5(vi), have been changed to $\mu=0$ , $\nu=0,1$ , and Addendum to §14.5(ii): $\mu=0$ , $\nu=2$ , respectively, in order to be more descriptive of their contents.
•

The second and the fourth lines of (19.7.2) containing $k^{\prime}/ik$ have both been replaced with $-ik^{\prime}/k$ to clarify the meaning.
•

Originally Equation (25.2.4) had the constraint $\Re s>0$ . This constraint was removed because, as stated after (25.2.1), $\zeta\left(s\right)$ is meromorphic with a simple pole at $s=1$ , and therefore $\zeta\left(s\right)-(s-1)^{-1}$ is an entire function. This was suggested by John Harper.
•

The title of §32.16 was changed from Physical to Physical Applications.
•

Bibliographic citations and clarifications have been added, removed, or modified in §§5.6(i), 5.10, 7.8, 7.25(iii), and 32.16.

… ►

Other Changes

•

To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

8.12.5 $\frac{{\mathrm{e}^{\pm\pi\mathrm{i}a}}}{2\mathrm{i}\sin(\pi a)}Q\left(-a,z{% \mathrm{e}^{\pm\pi\mathrm{i}}}\right)=\pm\tfrac{1}{2}\operatorname{erfc}\left(% \pm\mathrm{i}\eta\sqrt{a/2}\right)-\mathrm{i}T(a,\eta)$
•

Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_{k}=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}u_{k-1}$ was added to Equation (9.7.2).
•

In §15.2(ii), the unnumbered equation

$\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}% \left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1% )!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),$ $n=0,1,2,\dots$

was added in the second paragraph. An equation number will be assigned in an expanded numbering scheme that is under current development. Additionally, the discussion following (15.2.6) was expanded.
•

In §15.4(i), due to a report by Louis Klauder on 2017-01-01, and in §15.4(iii), sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula $F\left(a,b;a;z\right)=(1-z)^{-b}$ .
•

A bibliographic citation was added in §11.13(i).

… ►

Other Changes

•

Pochhammer symbols have been introduced in Equations (7.12.1), (7.12.2), (7.12.3), (7.12.4), (7.12.5), (25.5.7), (25.8.3), (25.11.10), (25.11.28), and (25.11.43) to make the notation more concise.
•

The Wronskian (14.2.7) was generalized to include both associated Legendre and Ferrers functions.
•

A cross-reference has been added in §15.9(iv).
•

Equations (22.19.6), (22.19.7), (22.19.8), and (22.19.9) have been rewritten with the modulus (second argument) of the Jacobian elliptic function defined explicitly in the preceding line of text.
•

Bibliographic citations have been added in §§4.13, 4.48(iv), 6.21(ii), 8.28(ii), 9.16, 10.77(viii), 12.21(ii), 14.28(ii), 14.34(ii), 16.4(ii) and 16.13.
•

An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).

…

27: 2.5 Mellin Transform Methods

… ►

2.5.9

\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)=\frac{\pi}{\sin\left(% \pi z\right)},

0<\Re z<1

ⓘ

Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $f(t)=1/(1+t)$ : function
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/2.5.E9
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

►

2.5.10

\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\frac{2^{z-1}\Gamma\left(% \nu+\frac{1}{2}z\right)}{{\Gamma^{2}}\left(1-\frac{1}{2}z\right)\Gamma\left(1+% \nu-\frac{1}{2}z\right)\Gamma\left(z\right)}\frac{\pi}{\sin\left(\pi z\right)},

-2\nu<\Re z<1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $h(t)={J_{\nu}^{2}}\left(t\right)$ : function and $f(t)=1/(1+t)$ : function
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/2.5.E10
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

… ►

2.5.11

\Residue_{z=n}\left[x^{-z}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z% \right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)\right]=(a_{n}\ln x% +b_{n})x^{-n},

ⓘ

Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $\ln\NVar{z}$ : principal branch of logarithm function, $\Residue$ : residue, $h(t)={J_{\nu}^{2}}\left(t\right)$ : function, $a_{n}$ : coefficients, $b_{n}$ : coefficients and $f(t)=1/(1+t)$ : function
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/2.5.E11
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

…

28: 1.14 Integral Transforms

… ►

1.14.17

\mathscr{L}\left(f\right)\left(s\right)=\mathscr{L}\mskip-3.0mu f\mskip 3.0mu % \left(s\right)=\int^{\infty}_{0}e^{-st}f(t)\mathrm{d}t.

ⓘ

Defines:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform
Symbols:: $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm and $\int$ : integral
Keywords:: Laplace transform
A&S Ref:: 29.1.1
Permalink:: http://dlmf.nist.gov/1.14.E17
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\left(f\right)\left(s\right)$ or $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

… ►

1.14.19

\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)\to 0,

\Re s\to\infty

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform and $\Re$ : real part
Keywords:: Laplace transform
Permalink:: http://dlmf.nist.gov/1.14.E19
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

… ►If also

\lim_{t\to 0+}f(t)/t

exists, then … ►where

A_{p}=\tan\left(\tfrac{1}{2}\pi/p\right)

when

1 <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math>, or <math class=

when

p\geq 2

. … ►

1.14.46

\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathcal{H}\mskip-3.0mu f\mskip 3.% 0mu \left(u\right)e^{iux}\mathrm{d}u=-\mathrm{i}(\operatorname{sign}x)\mathscr% {F}\mskip-3.0mu f\mskip 3.0mu \left(x\right),

ⓘ

Symbols:: $\mathscr{F}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Fourier transform, $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$ : Hilbert transform, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{sign}\NVar{x}$ : sign of $x$
Referenced by:: §1.14(v)
Permalink:: http://dlmf.nist.gov/1.14.E46
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Hilbert transform was changed to $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(t\right)$ from $\mathcal{H}(f;t)$ , and the notation for the Fourier transform was changed to $\mathscr{F}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ from $F(x)$ . In addition, the integration variable was changed from $t$ to $u$ .
See also:: Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

…

29: 15.12 Asymptotic Approximations

… ►

15.12.5

\mathbf{F}\left({a+\lambda,b-\lambda\atop c};\tfrac{1}{2}-\tfrac{1}{2}z\right)% =2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1)^{c/2}}\sqrt{\zeta\sinh\zeta}% \left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)^{1-c}\left(I_{c-1}\left((% \lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O(\lambda^{-2}))+\frac{I_{c% -2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)}{2\lambda+a-b}\left% (\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}\right)\left(\frac{1}{\zeta}-% \coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)\tanh\left(\tfrac{1}{2}\zeta% \right)+O(\lambda^{-2})\right)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.9

(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c% };-z\right)={\lambda^{-1/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(1/3))}\mathrm{% Ai}\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)+% e^{\pi\mathrm{i}(c-a-\lambda-(1/3))}\mathrm{Ai}\left(e^{\ifrac{2\pi\mathrm{i}}% {3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{0}(\zeta)+O(\lambda^% {-1})\right)}+\lambda^{-2/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(2/3))}\mathrm{% Ai}'\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)% +e^{\pi\mathrm{i}(c-a-\lambda-(2/3))}\mathrm{Ai}'\left(e^{\ifrac{2\pi\mathrm{i% }}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)+O(% \lambda^{-1})\right),

… ►

15.12.11

\beta=\left(-\frac{3}{2}\zeta+\frac{9}{4}\ln\left(\frac{2+e^{\zeta}}{2+e^{-% \zeta}}\right)\right)^{1/3},

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $\ln\NVar{z}$ : principal branch of logarithm function, $\zeta$ : change of variable and $\beta$
Permalink:: http://dlmf.nist.gov/15.12.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.13

G_{0}(\pm\beta)=\left(2+e^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}\left(1+e^{\pm% \zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-e^{\pm\zeta}\right)^{-a+(\ifrac{1}% {2})}\sqrt{\frac{\beta}{e^{\zeta}-e^{-\zeta}}}.

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\zeta$ : change of variable, $\beta$ and $G_{0}(\pm\zeta)$ : function
Permalink:: http://dlmf.nist.gov/15.12.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

…

30: 19.2 Definitions

… ►The cases with

\phi=\pi/2

are the complete integrals: … ►

k_{c}=k^{\prime},

►

p=1-\alpha^{2},

►

x=\tan\phi,

… ►If

1<k\leq 1/\sin\phi

, then

k_{c}

is pure imaginary. …