莱斯布里奇大学学历认证【购证 微kaa77788】big

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1: 22.10 Maclaurin Series

… ►

22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.2 $$\mathrm{cn}(z,k)=1-\frac{z^2}{2!}+\left(1+4k^2\right)\frac{z^4}{4!}-\left(1+44k^2+16k^4\right)\frac{z^6}{6!}+O\left(z^8\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.2

Permalink:

http://dlmf.nist.gov/22.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.3 $$\mathrm{dn}(z,k)=1-k^2\frac{z^2}{2!}+k^2\left(4+k^2\right)\frac{z^4}{4!}-k^2\left(16+44k^2+k^4\right)\frac{z^6}{6!}+O\left(z^8\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.3

Permalink:

http://dlmf.nist.gov/22.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

… ►

22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

… ►

22.10.6 $$\mathrm{dn}(z,k)=1-\frac{k^2}{2}{\sin }^{2}z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.3

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

…

2: 19.27 Asymptotic Approximations and Expansions

… ►

19.27.2 $$R_F(x,y,z)=\frac{1}{2\sqrt{z}}\left(\ln \frac{8z}{a+g}\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iii), §19.27(ii)

Permalink:

http://dlmf.nist.gov/19.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

►

19.27.3 $$R_F(x,y,z)=R_F(0,y,z)-\frac{1}{\sqrt{h}}\left(\sqrt{\frac{x}{h}}+O\left(\frac{x}{h}\right)\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

… ►

19.27.4 $$R_G(x,y,z)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln \frac{z}{a}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\ln z$: principal branch of logarithm function and $a$

Permalink:

http://dlmf.nist.gov/19.27.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

►

19.27.5 $$R_G(x,y,z)=R_G(0,y,z)+\sqrt{x}O\left(\sqrt{x/h}\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

►

19.27.6 $$R_G(0,y,z)=\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln \left(\frac{16z}{y}\right)-1\right)\left(1+O\left(\frac{y}{z}\right)\right),$$ $y/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/19.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

…

3: 27.11 Asymptotic Formulas: Partial Sums

… ►

27.11.1 $$\sum _{n\le x}f(n)=F(x)+O\left(g(x)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: 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 $$\sum _{n\le x}d\left(n\right)=x\ln x+(2\gamma -1)x+O\left(\sqrt{x}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: 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 $\log$.

See also:

Annotations for §27.11 and Ch.27

… ►Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number ${\theta }_0$ such that the error term in (27.11.2) is $O\left(x^{\theta }\right)$ for all $\theta >{\theta }_0$. … ►

27.11.3 $$\sum _{n\le x}\frac{d\left(n\right)}{n}=\frac{1}{2}{(\ln x)}^2+2\gamma \ln x+O\left(1\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: 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 $\log$.

See also:

Annotations for §27.11 and Ch.27

…

4: 25.9 Asymptotic Approximations

… ►

25.9.1 $$\zeta \left(\sigma +\mathrm{i}t\right)=\sum _{1\le n\le x}\frac{1}{n^s}+\chi (s)\sum _{1\le n\le y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma }\right)+O\left(y^{\sigma -1}{t}^{\frac{1}{2}-\sigma }\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable, $s$: complex variable and $\sigma$: fixed

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.12.4), p. 79)

Referenced by:

§25.9

Permalink:

http://dlmf.nist.gov/25.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

… ►

25.9.3 $$\zeta \left(\frac{1}{2}+\mathrm{i}t\right)=\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi \left(\frac{1}{2}+\mathrm{i}t\right)\sum _{n=1}^m\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+O\left(t^{-1/4}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

asymptotic approximation

Source:

Titchmarsh (1986b, (4.17.1), p. 88)

Referenced by:

§25.10(ii)

Permalink:

http://dlmf.nist.gov/25.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.9 and Ch.25

…

5: 2.1 Definitions and Elementary Properties

… ►The symbol $O$ can also apply to the whole set $𝐗$, and not just as $x\to c$. … ►The symbols $o$ and $O$ can be used generically. … ►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $𝐒$ and $f(z)=O\left(z^{\nu }\right)$ as $z\to \mathrm{\infty }$ in $𝐒$, $\nu$ being real, then $f^{\prime }(z)=O\left(z^{\nu -1}\right)$ as $z\to \mathrm{\infty }$ in any closed sector properly interior to $𝐒$ and with the same vertex (Ritt’s theorem). This result also holds with both $O$’s replaced by $o$’s. … ►Differentiation, however, requires the kind of extra conditions needed for the $O$ symbol (§2.1(ii)). …

6: 23.12 Asymptotic Approximations

… ►

23.12.1 $$\mathrm{\wp }\left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(-\frac{1}{3}+{\csc }^2\left(\frac{\pi z}{2{\omega }_1}\right)+8\left(1-\cos \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

►

23.12.2 $$\zeta \left(z\right)=\frac{{\pi }^2}{4{\omega }_1^2}\left(\frac{z}{3}+\frac{2{\omega }_1}{\pi }\cot \left(\frac{\pi z}{2{\omega }_1}\right)-8\left(z-\frac{{\omega }_1}{\pi }\sin \left(\frac{\pi z}{{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§23.12, Erratum (V1.0.23) for Equation (23.12.2)

Permalink:

http://dlmf.nist.gov/23.12.E2

Encodings:

pMML, png, TeX

Correction (effective with 1.0.23):

The factor of 2, previously omitted from the denominator of the argument of the $\cot$ function has been inserted.

Suggested 2019-05-27 by Blagoje Oblak

See also:

Annotations for §23.12 and Ch.23

►

23.12.3 $$\sigma \left(z\right)=\frac{2{\omega }_1}{\pi }\exp \left(\frac{{\pi }^2{z}^2}{24{\omega }_1^2}\right)\sin \left(\frac{\pi z}{2{\omega }_1}\right)\left(1-\left(\frac{{\pi }^2{z}^2}{{\omega }_1^2}-4{\sin }^2\left(\frac{\pi z}{2{\omega }_1}\right)\right)q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\sin z$: sine function, $𝕃$: lattice, $q$: nome, $z$: complex and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Permalink:

http://dlmf.nist.gov/23.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

… ►

23.12.4 $${\eta }_1=\frac{{\pi }^2}{4{\omega }_1}\left(\frac{1}{3}-8q^2+O\left(q^4\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $q$: nome, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and ${\eta }_j$: complex number

Referenced by:

§23.12

Permalink:

http://dlmf.nist.gov/23.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.12 and Ch.23

…

7: 27.18 Methods of Computation: Primes

… ►It runs in time $O\left({(\ln n)}^{c\ln \ln \ln n}\right)$. … ►That is to say, it runs in time $O\left({(\ln n)}^c\right)$ for some constant $c$. …

8: 3.4 Differentiation

… ►

3.4.20 $$\frac{\partial u_{0,0}}{\partial x}=\frac{1}{2h}(u_{1,0}-u_{-1,0})+O\left(h^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.21

Permalink:

http://dlmf.nist.gov/3.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

►

3.4.21 $$\frac{\partial u_{0,0}}{\partial x}=\frac{1}{4h}(u_{1,1}-u_{-1,1}+u_{1,-1}-u_{-1,-1})+O\left(h^2\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.22

Permalink:

http://dlmf.nist.gov/3.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►

3.4.22 $$\frac{{\partial }^2{u}_{0,0}}{{\partial x}^2}=\frac{1}{h^2}(u_{1,0}-2u_{0,0}+u_{-1,0})+O\left(h^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.23

Permalink:

http://dlmf.nist.gov/3.4.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►

3.4.25 $$\frac{{\partial }^2{u}_{0,0}}{\partial x\partial y}=\frac{1}{4h^2}(u_{1,1}-u_{1,-1}-u_{-1,1}+u_{-1,-1})+O\left(h^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\partial x$: partial differential of $x$ and $u$: function

A&S Ref:

25.3.26

Permalink:

http://dlmf.nist.gov/3.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

… ►

3.4.28 $${\nabla }^2{u}_{0,0}=\frac{1}{h^2}(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,-1}-4u_{0,0})+O\left(h^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.30

Permalink:

http://dlmf.nist.gov/3.4.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

…

9: 13.22 Zeros

… ►

13.22.1 $${\varphi }_r=\frac{j_{2\mu ,r}^2}{4\kappa }+j_{2\mu ,r}O\left({\kappa }^{-\frac{3}{2}}\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, ${\varphi }_r$: 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

…

10: 19.12 Asymptotic Approximations

… ►

19.12.4 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \frac{4}{k^{\prime }}\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right)-{\alpha }^2{R}_C(1,1-{\alpha }^2),$$ ,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

►

19.12.5 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \left(\frac{4}{k^{\prime }}\right)-R_C(1,1-{\alpha }^{-2})\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right),$$ .

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

… ►

19.12.6 $$R_C(x,y)=\frac{\pi }{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(\sqrt{\frac{x}{y}}\right)\right),$$ $x/y\to 0$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\pi$: the ratio of the circumference of a circle to its diameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

►

19.12.7 $$R_C(x,y)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)(1+O\left(y^2/x^2\right)),$$ $y/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln z$: principal branch of logarithm function

Referenced by:

§19.12, §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19