DLMF: Untitled Document

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Abramowitz and Stegun (1964, Chapter 7) includes $\operatorname{erf}x$ , $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[0,2]$ , 10D; $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[2,10]$ , 8S; $xe^{x^{2}}\operatorname{erfc}x$ , $x^{-2}\in[0,0.25]$ , 7D; $2^{n}\Gamma\left(\frac{1}{2}n+1\right)\mathop{\mathrm{i}^{n}\mathrm{erfc}}% \left(x\right)$ , $n=1(1)6,10,11$ , $x\in[0,5]$ , 6S; $F\left(x\right)$ , $x\in[0,2]$ , 10D; $xF\left(x\right)$ , $x^{-2}\in[0,0.25]$ , 9D; $C\left(x\right)$ , $S\left(x\right)$ , $x\in[0,5]$ , 7D; $\mathrm{f}\left(x\right)$ , $\mathrm{g}\left(x\right)$ , $x\in[0,1]$ , $x^{-1}\in[0,1]$ , 15D.

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Finn and Mugglestone (1965) includes the Voigt function $H\left(a,u\right)$ , $u\in[0,22]$ , $a\in[0,1]$ , 6S.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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… ►

P^{\mu}_{\nu}\left(x\pm i0\right)

(either side of the cut) has exactly one zero in the interval

(-\infty,-1)

if either of the following sets of conditions holds: …For all other values of the parameters

P^{\mu}_{\nu}\left(x\pm i0\right)

has no zeros in the interval

(-\infty,-1)

. …

… ►If

(j,k)\in B

, then

\sigma(j)\neq k

. The number of derangements of

n

is the number of permutations with forbidden positions

B=\{(1,1),(2,2),\ldots,(n,n)\}

. … ►For

(j,k)\in B

,

B\setminus[j,k]

denotes

B

after removal of all elements of the form

(j,t)

or

(t,k)

,

t=1,2,\ldots,n

.

B\setminus(j,k)

denotes

B

with the element

(j,k)

removed. … ►Let

B=\{(j,j),(j,j+1)\>|\>1\leq j<n\}\cup\{(n,n),(n,1)\}

. …

… ►

§14.16(ii) Interval $-1<x<1$

… ►The zeros of

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

in the interval

(-1,1)

interlace those of

\mathsf{P}^{\mu}_{\nu}\left(x\right)

. … ►

§14.16(iii) Interval $1<x<\infty$

►

P^{\mu}_{\nu}\left(x\right)

has exactly one zero in the interval

(1,\infty)

if either of the following sets of conditions holds: … ►

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)

has no zeros in the interval

(1,\infty)

when

\nu>-1

, and at most one zero in the interval

(1,\infty)

when

\nu<-1

.

§22.17 Moduli Outside the Interval [0,1]

… ►Jacobian elliptic functions with real moduli in the intervals

(-\infty,0)

and

(1,\infty)

, or with purely imaginary moduli are related to functions with moduli in the interval

[0,1]

by the following formulas. … ►For proofs of these results and further information see Walker (2003).

… ► … ►Suppose

f(x)

is defined on

[a,b]

. … ►Then for

f(x)

continuous on

(a,b)

, … ►for any

c,d\in(a,b)

, and

t\in[0,1]

. …A similar definition applies to closed intervals

[a,b]

. …

… ►Let

x^{\prime}\in(a,b)

. … ►Here

x(t,N)

is an interpolation of the abscissas

x_{i,N},i=1,2,\dots,N

, that is,

x(i,N)=x_{i,N}

, allowing differentiation by

i

. …The PWCF

x(t,N)

is a minimally oscillatory algebraic interpolation of the abscissas

x_{i,N},i=1,2,\dots,N

. … ►This is a challenging case as the desired

w^{\mathrm{RCP}}(x)

on

[-1,1]

has an essential singularity at

x=-1

. … ►Further, exponential convergence in

N

, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate

w(x)

for these OP systems on

x\in[-1,1]

and

(-\infty,\infty)

respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

… ►

D(m,n)

is the number of paths from

(0,0)

to

(m,n)

that are composed of directed line segments of the form

(1,0)

,

(0,1)

, or

(1,1)

. … ►

M(n)

is the number of lattice paths from

(0,0)

to

(n,n)

that stay on or above the line

y=x

and are composed of directed line segments of the form

(2,0)

,

(0,2)

, or

(1,1)

. … ►

N(n,k)

is the number of lattice paths from

(0,0)

to

(n,n)

that stay on or above the line

y=x

, are composed of directed line segments of the form

(1,0)

or

(0,1)

, and for which there are exactly

k

occurrences at which a segment of the form

(0,1)

is followed by a segment of the form

(1,0)

. … ►

r(n)

is the number of paths from

(0,0)

to

(n,n)

that stay on or above the diagonal

y=x

and are composed of directed line segments of the form

(1,0)

,

(0,1)

, or

(1,1)

. …

… ►In (4.37.2) the integration path may not pass through either of the points

\pm 1

, and the function

(t^{2}-1)^{1/2}

assumes its principal value when

t\in(1,\infty)

. … ►

4.37.19

\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),

z\in\mathbb{C}\setminus(-\infty,1)

,

ⓘ

Symbols:: $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction and $z$ : complex variable
Referenced by:: §4.37(iv), §4.37(iii), §4.37(iv)
Permalink:: http://dlmf.nist.gov/4.37.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

… ►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on

(-\infty,1]

. … ►

4.37.22

\operatorname{arccosh}x=\pm\ln\left(i(1-x^{2})^{1/2}+x\right),

x\in(-1,1]

,

ⓘ

Symbols:: $\in$ : element of, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.37.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

… ►

4.37.24

\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),

z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)

;

ⓘ

Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\in$ : element of, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.37.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

…

… ►Let

\theta_{n,m}=\theta_{n,m}^{(\alpha,\beta)}

,

m=1,2,\dots,n

, denote the zeros of

P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)

as function of

\theta

with … ►

18.16.2

\theta_{n,m}^{(-\frac{1}{2},\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{% 1}{2}}\leq\theta_{n,m}^{(\alpha,\beta)}\leq\frac{m\pi}{n+\tfrac{1}{2}}=\theta_% {n,m}^{(\frac{1}{2},-\frac{1}{2})},

\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta_{n,m}$ : zeros
Referenced by:: §18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3)
Permalink:: http://dlmf.nist.gov/18.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

►

18.16.3

\theta_{n,m}^{(-\frac{1}{2},-\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n}\leq% \theta_{n,m}^{(\alpha,\alpha)}\leq\frac{m\pi}{n+1}=\theta_{n,m}^{(\frac{1}{2},% \frac{1}{2})},

\alpha\in[-\tfrac{1}{2},\tfrac{1}{2}]

,

m=1,2,\dots,\left\lfloor\frac{1}{2}n\right\rfloor

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $(\NVar{a},\NVar{b})$ : open interval, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta_{n,m}$ : zeros
Referenced by:: §18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3)
Permalink:: http://dlmf.nist.gov/18.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

… ►when

\alpha\notin(-\frac{1}{2},\frac{1}{2})

. … ►All zeros of

H_{n}\left(x\right)

lie in the open interval

(-\sqrt{2n+1},\sqrt{2n+1})

. …

of interval

1—10 of 237 matching pages

1: 7.23 Tables

2: 14.27 Zeros

3: 26.15 Permutations: Matrix Notation

4: 14.16 Zeros

§14.16(ii) Interval $-1<x<1$

§14.16(iii) Interval $1<x<\infty$

5: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

6: 1.4 Calculus of One Variable

7: 18.40 Methods of Computation

8: 26.6 Other Lattice Path Numbers

9: 4.37 Inverse Hyperbolic Functions

10: 18.16 Zeros

of interval

1—10 of 237 matching pages

1: 7.23 Tables

2: 14.27 Zeros

3: 26.15 Permutations: Matrix Notation

4: 14.16 Zeros

§14.16(ii) Interval − 1 < x < 1

§14.16(iii) Interval 1 < x < ∞

5: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

6: 1.4 Calculus of One Variable

7: 18.40 Methods of Computation

8: 26.6 Other Lattice Path Numbers

9: 4.37 Inverse Hyperbolic Functions

10: 18.16 Zeros

§14.16(ii) Interval $-1<x<1$

§14.16(iii) Interval $1<x<\infty$