small x

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41—50 of 102 matching pages

41: 28.35 Tables

… ►

•

Blanch and Clemm (1965) includes values of ${\operatorname{Mc}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Mc}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=0(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . Also ${\operatorname{Ms}^{(2)}_{n}}\left(x,\sqrt{q}\right)$ , ${\operatorname{Ms}^{(2)}_{n}}'\left(x,\sqrt{q}\right)$ for $n=1(1)7$ , $x=0(.02)1$ ; $n=8(1)15$ , $x=0(.01)1$ . In all cases $q=0(.05)1$ . Precision is generally 7D. Approximate formulas and graphs are also included.

… ►

•

Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$ , $\operatorname{Se}_{n+1}\left(x,q\right)$ for $n=0(1)5$ , $q=1(1)20$ , $x=0.1(.1)1$ ; 4D or 5D.

… ►

•

Stratton et al. (1941) includes $b_{n}$ , $b_{n}^{\prime}$ , and the corresponding Fourier coefficients for $\mathrm{Se}_{n}(c,x)$ and $\mathrm{So}_{n}(c,x)$ for $n=0$ or $1(1)4$ , $c=0(.1~{}\textrm{or}~{}.2)4.5$ . Precision is mostly 5S. Notation: $c=2\sqrt{q}$ , $b_{n}=a_{n}+2q$ , $b^{\prime}_{n}=b_{n}+2q$ , and for $\mathrm{Se}_{n}(c,x)$ , $\mathrm{So}_{n}(c,x)$ see §28.1.

►

•

Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n+1}\left(q\right)$ for $n=0(1)4$ , $q=0(1)50$ ; $n=0(1)20$ ( $a$ ’s) or 19 ( $b$ ’s), $q=1,3,5,10,15,25,50(50)200$ . Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , $n=0(1)7$ . Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $x=0(5^{\circ})90^{\circ}$ . Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $j=1,2$ , $x=0(.2)4$ . Precision is mostly 9S.

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•

Ince (1932) includes the first zero for $\operatorname{ce}_{n}$ , $\operatorname{se}_{n}$ for $n=2(1)5$ or $6$ , $q=0(1)10(2)40$ ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small $q$ .

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42: 1.4 Calculus of One Variable

… ►that is, for every arbitrarily small positive constant

\epsilon

there exists

\delta

(

>0

) such that … ►The derivative

f^{\prime}(x)

f(x)

is defined by … ►For

h(x)=f(g(x))

, … ►For

F^{\prime}(x)=f(x)

with

f(x)

continuous, … ►This definition also applies when

f(x)

is a complex function of the real variable

x

. …

43: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

… ►uniformly for

x\in(0,1)

and

a/(a+b)

b/(a+b)\in[\delta,1-\delta]

, where

\delta

again denotes an arbitrary small positive constant. …

44: 12.1 Special Notation

… ► ►►►

$x, y$	real variables.
…
$\delta$	arbitrary small positive constant.

…

45: 11.6 Asymptotic Expansions

… ►

11.6.3

\int_{0}^{z}\mathbf{K}_{0}\left(t\right)\,\mathrm{d}t-\frac{2}{\pi}(\ln\left(2% z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(2k)!(2k-1% )!}{(k!)^{2}(2z)^{2k}},

|\operatorname{ph}z|\leq\pi-\delta

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.1.32
Referenced by:: §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

►

11.6.4

\int_{0}^{z}\mathbf{M}_{0}\left(t\right)\,\mathrm{d}t+\frac{2}{\pi}(\ln\left(2% z\right)+\gamma)\sim\frac{2}{\pi}\sum_{k=1}^{\infty}\frac{(2k)!(2k-1)!}{(k!)^{% 2}(2z)^{2k}},

|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $k$ : nonnegative integer and $\delta$ : arbitrary small positive constant
A&S Ref:: 12.2.8
Referenced by:: §11.6(i)
Permalink:: http://dlmf.nist.gov/11.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(i), §11.6 and Ch.11

…

46: 8.1 Special Notation

… ► ►►►

$x$	real variable.
…
$\delta$	arbitrary small positive constant.
…

… ►The functions treated in this chapter are the incomplete gamma functions

\gamma\left(a,z\right)

\Gamma\left(a,z\right)

\gamma^{*}\left(a,z\right)

P\left(a,z\right)

, and

Q\left(a,z\right)

; the incomplete beta functions

\mathrm{B}_{x}\left(a,b\right)

and

I_{x}\left(a,b\right)

; the generalized exponential integral

E_{p}\left(z\right)

; the generalized sine and cosine integrals

\operatorname{si}\left(a,z\right)

\operatorname{ci}\left(a,z\right)

\operatorname{Si}\left(a,z\right)

, and

\operatorname{Ci}\left(a,z\right)

. ►Alternative notations include: Prym’s functions

P_{z}(a)=\gamma\left(a,z\right)

Q_{z}(a)=\Gamma\left(a,z\right)

, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63);

(a,z)!=\gamma\left(a+1,z\right)

[a,z]!=\Gamma\left(a+1,z\right)

, Dingle (1973);

B(a,b,x)=\mathrm{B}_{x}\left(a,b\right)

I(a,b,x)=I_{x}\left(a,b\right)

, Magnus et al. (1966);

\operatorname{Si}\left(a,x\right)\to\operatorname{Si}\left(1-a,x\right)

\operatorname{Ci}\left(a,x\right)\to\operatorname{Ci}\left(1-a,x\right)

, Luke (1975).

47: 30.1 Special Notation

… ► ►►►

$x$	real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, $-1<x<1$ .
…
$\delta$	arbitrary small positive constant.

►The main functions treated in this chapter are the eigenvalues

\lambda^{m}_{n}\left(\gamma^{2}\right)

and the spheroidal wave functions

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)

\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right)

\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right)

, and

S^{m(j)}_{n}\left(z,\gamma\right)

j=1,2,3,4

. …Meixner and Schäfke (1954) use

\mathrm{ps}

\mathrm{qs}

\mathrm{Ps}

\mathrm{Qs}

for

\mathsf{Ps}

\mathsf{Qs}

\mathit{Ps}

\mathit{Qs}

, respectively. … ►Flammer (1957) and Abramowitz and Stegun (1964) use

\lambda_{mn}(\gamma)

for

\lambda^{m}_{n}\left(\gamma^{2}\right)+\gamma^{2}

R_{mn}^{(j)}(\gamma,z)

for

S^{m(j)}_{n}\left(z,\gamma\right)

, and …

48: 28.1 Special Notation

… ► ►►►►►

$m, n$	integers.
$x, y$	real variables.
$z=x+\mathrm{i}y$	complex variable.
…
$\delta$	arbitrary small positive number.
…

…

49: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►Throughout this section the symbol

\delta

again denotes an arbitrary small positive constant. … ►

§12.10(iii) Negative $a$ , $-\infty<x<-2\sqrt{-a}$

… ►

§12.10(iv) Negative $a$ , $-2\sqrt{-a}<x<2\sqrt{-a}$

… ►

§12.10(v) Positive $a$ , $-\infty<x<\infty$

… ►

§12.10(viii) Negative $a$ , $-\infty<x<2\sqrt{-a}$ . Expansions in Terms of Airy Functions

…

50: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►uniformly with respect to

x\in(0,\infty)

and

\kappa\in[0,(1-\delta)\mu]

, where

\delta

again denotes an arbitrary small positive constant. …

small x

41—50 of 102 matching pages

41: 28.35 Tables

42: 1.4 Calculus of One Variable

43: 8.18 Asymptotic Expansions of I x ⁡ ( a , b )

44: 12.1 Special Notation

45: 11.6 Asymptotic Expansions

46: 8.1 Special Notation

47: 30.1 Special Notation

48: 28.1 Special Notation

49: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10(iii) Negative a , − ∞ < x < − 2 ⁢ − a

§12.10(iv) Negative a , − 2 ⁢ − a < x < 2 ⁢ − a

§12.10(v) Positive a , − ∞ < x < ∞

§12.10(viii) Negative a , − ∞ < x < 2 ⁢ − a . Expansions in Terms of Airy Functions

50: 13.20 Uniform Asymptotic Approximations for Large μ

43: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

§12.10(iii) Negative $a$ , $-\infty<x<-2\sqrt{-a}$

§12.10(iv) Negative $a$ , $-2\sqrt{-a}<x<2\sqrt{-a}$

§12.10(v) Positive $a$ , $-\infty<x<\infty$

§12.10(viii) Negative $a$ , $-\infty<x<2\sqrt{-a}$ . Expansions in Terms of Airy Functions

50: 13.20 Uniform Asymptotic Approximations for Large $\mu$