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21: 3.8 Nonlinear Equations

… ►This iterative method for solving $z=\varphi (z)$ is given by … ► $p(z)=z^4-1$. … ► $p(z)=z^4-2z^2+1$. … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

22: 32.8 Rational Solutions

… ►where the $Q_n(z)$ are monic polynomials (coefficient of highest power of $z$ is $1$) satisfying …with $Q_0(z)=1$, $Q_1(z)=z$. … ►Next, let $p_m(z)$ be the polynomials defined by $p_m(z)=0$ for , and …where ${\tau }_n(z)$ is the $n\times n$ Wronskian determinant … ►where $P_m(z)$ and $Q_m(z)$ are polynomials of degree $m$, with no common zeros. …

23: 28.1 Special Notation

… ► ►►►

$m,n$	integers.
…
$z=x+\mathrm{i}y$	complex variable.
…

… ► ►►

${\mathrm{ce}}_{\nu }(z,q)$,

${\mathrm{se}}_{\nu }(z,q)$,

${\mathrm{fe}}_n(z,q)$,

${\mathrm{ge}}_n(z,q)$,

${\mathrm{me}}_{\nu }(z,q)$,

… ►The functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are also known as the radial Mathieu functions. … ►

Abramowitz and Stegun (1964, Chapter 20)

… ►The radial functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are denoted by ${\mathrm{Mc}}_n^{(j)}(z,q)$ and ${\mathrm{Ms}}_n^{(j)}(z,q)$, respectively.

24: 36.5 Stokes Sets

… ►The Stokes set takes different forms for $z=0$, , and $z>0$. ►For $z=0$, the set consists of the two curves … ►For $z\ne 0$, the Stokes set is expressed in terms of scaled coordinates … ►For , there are two solutions $u$, provided that $|Y|>{(\frac{2}{5})}^{1/2}$. … ►For $z>0$ the Stokes set has two sheets. …

25: 6.16 Mathematical Applications

… ►

6.16.1

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/6.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

ⓘ

Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

6.16.3 $$R_n(x)=\frac{1}{2}{\int }_0^x\left(\frac{1}{\sin t}-\frac{1}{t}\right)\sin \left(2nt\right)dt.$$

ⓘ

Defines:

$R_n(x)$: remainder term (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

6.16.5 $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\ln x\right),$$ $x\to \mathrm{\infty }$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $x$: real variable and $\pi (x)$: number of primes $\le x$

A&S Ref:

5.1.50

Permalink:

http://dlmf.nist.gov/6.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(ii), §6.16 and Ch.6

… ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

26: 10.3 Graphics

… ►

►

►

Figure 10.3.14: $H_5^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

… ►

►

►

Figure 10.3.16: $H_{5.5}^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

…

27: 3.4 Differentiation

… ►

$$B_{-2}^5=\frac{1}{120}(6-10t-15t^2+20t^3-5t^4),$$

… ►

$$B_{-3}^6=-\frac{1}{720}(12-8t-45t^2+20t^3+15t^4-6t^5),$$

►

$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

… ►

$$B_2^6=-\frac{1}{60}(9+9t-30t^2-20t^3+5t^4+3t^5),$$

… ► $f(z)={\mathrm{e}}^z$, $x_0=0$. …

28: 12.19 Tables

… ►

Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U(n-\frac{1}{2},x)\right)$ for $n=1(1)20$, $x=0(.05)3$, 7S.

►

Zhang and Jin (1996, pp. 455–473) includes $U(\pm n-\frac{1}{2},x)$, $V(\pm n-\frac{1}{2},x)$, $U(\pm \nu -\frac{1}{2},x)$, $V(\pm \nu -\frac{1}{2},x)$, and derivatives, $\nu =n+\frac{1}{2}$, $n=0(1)10(10)30$, $x=0.5,1,5,10,30,50$, 8S; $W(a,\pm x)$, $W(-a,\pm x)$, and derivatives, $a=h(1)5+h$, $x=0.5,1$ and $a=h(1)5+h$, $x=5$, $h=0,0.5$, 8S. Also, first zeros of $U(a,x)$, $V(a,x)$, and of derivatives, $a=-6(.5)-1$, 6D; first three zeros of $W(a,-x)$ and of derivative, $a=0(.5)4$, 6D; first three zeros of $W(-a,\pm x)$ and of derivative, $a=0.5(.5)5.5$, 6D; real and imaginary parts of $U(a,z)$, $a=-1.5(1)1.5$, $z=x+\mathrm{i}y$, $x=0.5,1,5,10$, $y=0(.5)10$, 8S.

…

29: 36.4 Bifurcation Sets

… ►

$z\le 0,$

►

$x=\frac{9}{20}{z}^2.$

… ►

$x=-\frac{3}{20}{z}^2,$

… ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid … ►The $+$ sign labels the cusped sheet; the $-$ sign labels the sheet that is smooth for $z\ne 0$ (see Figure 36.4.4). …

30: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …