large order

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41—50 of 89 matching pages

41: 13.22 Zeros

… ►Asymptotic approximations to the zeros when the parameters

\kappa

and/or

\mu

are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if

\mu(\geq 0)

is fixed and

\kappa(>0)

is large, then the

r

th positive zero

\phi_{r}

M_{\kappa,\mu}\left(z\right)

is given by ►

13.22.1

\phi_{r}=\frac{j_{2\mu,r}^{2}}{4\kappa}+j_{2\mu,r}O\left(\kappa^{-\frac{3}{2}}% \right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi_{r}$ : positive zeros and $j_{b,r}$ : positive zero of Bessel
Referenced by:: §13.22, §13.22
Permalink:: http://dlmf.nist.gov/13.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.22 and Ch.13

…

42: 2.4 Contour Integrals

… ►Except that

\lambda

is now permitted to be complex, with

\Re\lambda>0

, we assume the same conditions on

q(t)

and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of

\Re z

. Then … ►For large

t

, the asymptotic expansion of

q(t)

may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function

F(z)

for

Q(z)

that has an inverse transform …If this integral converges uniformly at each limit for all sufficiently large

t

, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►in which

z

is a large real or complex parameter,

p(\alpha,t)

and

q(\alpha,t)

are analytic functions of

t

and continuous in

t

and a second parameter

\alpha

. …

43: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

… ►

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

… ►

§13.8(iii) Large $a$

… ► … ►

§13.8(iv) Large $a$ and $b$

…

44: 16.11 Asymptotic Expansions

… ►

16.11.10

{{}_{p+1}F_{p}}\left({a_{1}+r,\dots,a_{k-1}+r,a_{k},\dots,a_{p+1}\atop b_{1}+r% ,\dots,b_{k}+r,b_{k+1},\dots,b_{p}};z\right)=\sum_{n=0}^{m-1}\frac{{\left(a_{1% }+r\right)_{n}}\cdots{\left(a_{k-1}+r\right)_{n}}{\left(a_{k}\right)_{n}}% \cdots{\left(a_{p+1}\right)_{n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{k}% +r\right)_{n}}{\left(b_{k+1}\right)_{n}}\cdots{\left(b_{p}\right)_{n}}}\frac{z% ^{n}}{n!}+O\left(\frac{1}{r^{m}}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $r$ : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(iii), §16.11 and Ch.16

… ►

16.11.11

{{}_{p}F_{q}}\left({a_{1}+r,\dots,a_{p}+r\atop b_{1}+r,\dots,b_{q}+r};z\right)% =\sum_{n=0}^{m-1}\frac{{\left(a_{1}+r\right)_{n}}\cdots{\left(a_{p}+r\right)_{% n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{q}+r\right)_{n}}}\frac{z^{n}}{n% !}+O\left(\frac{1}{r^{(q-p)m}}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $r$ : large parameter
Permalink:: http://dlmf.nist.gov/16.11.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §16.11(iii), §16.11 and Ch.16

…

45: 2.8 Differential Equations with a Parameter

… ►

2.8.15

W_{n,1}(u,\xi)=\operatorname{Ai}\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-1}% \frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+% \operatorname{Ai}'\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi% )}{u^{2s+(4/3)}}+O\left(\frac{1}{u^{2n-1}}\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{n,j}(u,\xi)$ : solution, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Referenced by:: §2.1(iii), §2.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/2.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iii), §2.8 and Ch.2

►

2.8.16

W_{n,2}(u,\xi)=\operatorname{Bi}\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-1}% \frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+% \operatorname{Bi}'\left(u^{2/3}\xi\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi% )}{u^{2s+(4/3)}}+O\left(\frac{1}{u^{2n-1}}\right)\right).

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{n,j}(u,\xi)$ : solution, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Referenced by:: §2.8(iii), §2.8(iii)
Permalink:: http://dlmf.nist.gov/2.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iii), §2.8 and Ch.2

… ►

2.8.22

W_{n,1}(u,\xi)=\operatorname{Ai}\left(u^{2/3}\xi\right)\sum_{s=0}^{n-1}\frac{A% _{s}(\xi)}{u^{2s}}+\operatorname{Ai}'\left(u^{2/3}\xi\right)\sum_{s=0}^{n-2}% \frac{B_{s}(\xi)}{u^{2s+(4/3)}}+\operatorname{envAi}\left(u^{2/3}\xi\right)O% \left(\frac{1}{u^{2n-1}}\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{envAi}\left(\NVar{x}\right)$ : envelope of Airy function $\operatorname{Ai}\left(\NVar{x}\right)$ , $W_{n,j}(u,\xi)$ : solution, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Permalink:: http://dlmf.nist.gov/2.8.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iii), §2.8 and Ch.2

… ►

2.8.29

W_{n,3}(u,\xi)=|\xi|^{1/2}J_{\nu}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-% 1}\frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi|J_{% \nu+1}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+1% }}+O\left(\frac{1}{u^{2n-2}}\right)\right),

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{n,j}(u,\xi)$ : solution, $\nu$ : real nonnegative constant, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Referenced by:: §2.8(iv), §2.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.E29
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iv), §2.8 and Ch.2

►

2.8.30

W_{n,4}(u,\xi)=|\xi|^{1/2}Y_{\nu}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-% 1}\frac{A_{s}(\xi)}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)-|\xi|Y_{% \nu+1}\left(u|\xi|^{1/2}\right)\left(\sum_{s=0}^{n-2}\frac{B_{s}(\xi)}{u^{2s+1% }}+O\left(\frac{1}{u^{2n-2}}\right)\right).

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $O\left(\NVar{x}\right)$ : order not exceeding, $W_{n,j}(u,\xi)$ : solution, $\nu$ : real nonnegative constant, $n$ : positive integer, $A_{s}(\xi)$ : coefficients, $B_{s}(\xi)$ : coefficients and $u$ : large real or complex parameter
Referenced by:: §2.8(iv), §2.8(iv)
Permalink:: http://dlmf.nist.gov/2.8.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §2.8(iv), §2.8 and Ch.2

…

46: 10.68 Modulus and Phase Functions

… ►

§10.68(iii) Asymptotic Expansions for Large Argument

… ►

10.68.16

M_{\nu}\left(x\right)=\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\left(1-% \frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)^{2}}{256}\frac{1}{x^{2}}-% \frac{(\mu-1)(\mu^{2}+14\mu-399)}{6144\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}% {x^{4}}\right)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.21
Permalink:: http://dlmf.nist.gov/10.68.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

►

10.68.17

\ln M_{\nu}\left(x\right)=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln\left(2\pi x\right)% -\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1% }{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x^{5}}% \right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $M_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of Bessel functions, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.22
Permalink:: http://dlmf.nist.gov/10.68.E17
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

►

10.68.18

\theta_{\nu}\left(x\right)=\frac{x}{\sqrt{2}}+\left(\frac{1}{2}\nu-\frac{1}{8}% \right)\pi+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{\mu-1}{16}\frac{1}{x^{2}}-% \frac{(\mu-1)(\mu-25)}{384\sqrt{2}}\frac{1}{x^{3}}+O\left(\frac{1}{x^{5}}% \right).

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\theta_{\NVar{\nu}}\left(\NVar{x}\right)$ : phase of Bessel functions, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.23
Referenced by:: §10.68(i), §10.70
Permalink:: http://dlmf.nist.gov/10.68.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

… ►

10.68.20

\ln N_{\nu}\left(x\right)=-\frac{x}{\sqrt{2}}+\frac{1}{2}\ln\left(\frac{\pi}{2% x}\right)+\frac{\mu-1}{8\sqrt{2}}\frac{1}{x}+\frac{(\mu-1)(\mu-25)}{384\sqrt{2% }}\frac{1}{x^{3}}-\frac{(\mu-1)(\mu-13)}{128}\frac{1}{x^{4}}+O\left(\frac{1}{x% ^{5}}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $N_{\NVar{\nu}}\left(\NVar{x}\right)$ : modulus of derivatives of Bessel functions, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.10.25
Permalink:: http://dlmf.nist.gov/10.68.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §10.68(iii), §10.68 and Ch.10

…

47: 30.11 Radial Spheroidal Wave Functions

… ►

§30.11(iii) Asymptotic Behavior

… ►

30.11.6

S^{m(j)}_{n}\left(z,\gamma\right)=\begin{cases}\psi_{n}^{(j)}(\gamma z)+O\left% (z^{-2}e^{|\Im z|}\right),&j=1,2,\\ \psi_{n}^{(j)}(\gamma z)\left(1+O\left(z^{-1}\right)\right),&j=3,4.\end{cases}

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\psi_{k}^{(j)}(z)$ : spherical function and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §30.11(iii), §30.11 and Ch.30

…

48: 25.11 Hurwitz Zeta Function

… ►

25.11.28

\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}% \frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+\frac{1}{\Gamma\left(s% \right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_% {k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-ax}\,\mathrm{d}x,

\Re s>-(2n+1)

s\neq 1

\Re a>0

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Keywords:: improper integral, integral representation
Proof sketch:: Derivable from (25.11.27) by adding and subtracting $\sum_{k=1}^{n}B_{2k}x^{2k-1}/(2k)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that $({\mathrm{e}}^{x}-1)^{-1}-x^{-1}+1/2-\sum_{k=1}^{n}B_{2k}x^{2k-1}/(2k)!=O\left% (x^{2n+1}\right)$ as $x\to 0$ , demonstrating the region of convergence.
Referenced by:: Erratum (V1.0.9) for Chapters 7, 25
Permalink:: http://dlmf.nist.gov/25.11.E28
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.9):: We have rewritten the original summation $\sum_{k=1}^{n}\frac{\Gamma\left(s+2k-1\right)}{\Gamma\left(s\right)}\frac{B_{2% k}}{(2k)!}a^{-2k-s+1}$ more concisely as $\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}$ using the Pochhammer symbol.
See also:: Annotations for §25.11(vii), §25.11 and Ch.25

… ►

§25.11(xii) $a$ -Asymptotic Behavior

… ►

25.11.41

\zeta\left(s,a+1\right)=\zeta\left(s\right)-s\zeta\left(s+1\right)a+O\left(a^{% 2}\right).

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $a$ : real or complex parameter and $s$ : complex variable
Keywords:: asymptotic approximation
Source:: Apostol (1952, (15), p. 6)
Permalink:: http://dlmf.nist.gov/25.11.E41
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(xii), §25.11 and Ch.25

… ►Similarly, as

a\to\infty

in the sector

|\operatorname{ph}a|\leq\frac{1}{2}\pi-\delta(<\frac{1}{2}\pi)

, …

49: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

►

§33.10(i) Large $\rho$

… ►

§33.10(ii) Large Positive $\eta$

… ►

§33.10(iii) Large Negative $\eta$

…

50: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function

f(n)

for large

n

is often difficult to determine because the function values can fluctuate considerably as

n

increases. … ►

27.11.1

\sum_{n\leq x}f(n)=F(x)+O\left(g(x)\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $n$ : positive integer and $x$ : real number
Permalink:: http://dlmf.nist.gov/27.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §27.11 and Ch.27

►where

F(x)

is a known function of

x

, and

O\left(g(x)\right)

represents the error, a function of smaller order than

F(x)

for all

x

in some prescribed range. … ►

27.11.2

\sum_{n\leq x}d\left(n\right)=x\ln x+(2\gamma-1)x+O\left(\sqrt{x}\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
A&S Ref:: 24.3.3 III
Referenced by:: §27.11, §27.11, Erratum (V1.1.8) for Section 27.11
Permalink:: http://dlmf.nist.gov/27.11.E2
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 §27.11 and Ch.27

… ►

27.11.3

\sum_{n\leq x}\frac{d\left(n\right)}{n}=\frac{1}{2}(\ln x)^{2}+2\gamma\ln x+O% \left(1\right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
Referenced by:: §27.11
Permalink:: http://dlmf.nist.gov/27.11.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 §27.11 and Ch.27

…