Visit (-- RXLARA.COM --) pharmacy buy Silagra over counter. Sildenafil Citrate Silagra Cipla 100mg pills price prescription onlin

SearchAdvancedHelp

(0.008 seconds)

11—20 of 335 matching pages

11: 26.15 Permutations: Matrix Notation

… ►where the sum is over and $n\ge h>\mathrm{\ell }\ge 1$. … ►For $(j,k)\in B$, $B\setminus [j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\mathrm{\dots },n$. $B\setminus (j,k)$ denotes $B$ with the element $(j,k)$ removed. ►

26.15.5 $$R(x,B)=xR(x,B\setminus [j,k])+R(x,B\setminus (j,k)).$$

ⓘ

Symbols:

$[a,b]$: closed interval, $(a,b)$: open interval, $\setminus$: set subtraction, $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $B$: subset and $R(x,B)$: rook polynomial

Permalink:

http://dlmf.nist.gov/26.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

…

12: 33.8 Continued Fractions

… ►

33.8.2 $$\frac{H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }}{H_{\mathrm{\ell }}^{\pm }}=c\pm \frac{\mathrm{i}}{\rho }\frac{ab}{2(\rho -\eta \pm \mathrm{i})+}\frac{(a+1)(b+1)}{2(\rho -\eta \pm 2\mathrm{i})+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient, $b$: coefficient and $c$: coefficient

Referenced by:

§33.8

Permalink:

http://dlmf.nist.gov/33.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

… ►

$a=1+\mathrm{\ell }\pm \mathrm{i}\eta ,$

►

$b=-\mathrm{\ell }\pm \mathrm{i}\eta ,$

►

$c=\pm \mathrm{i}(1-(\eta /\rho )).$

… ►

$F_{\mathrm{\ell }}=\pm {(q^{-1}{(u-p)}^2+q)}^{-1/2},$

…

13: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.13 $$\mathrm{Arcsin}u\pm \mathrm{Arcsin}v=\mathrm{Arcsin}\left(u{(1-v^2)}^{1/2}\pm v{(1-u^2)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.32

Permalink:

http://dlmf.nist.gov/4.24.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.15 $$\mathrm{Arctan}u\pm \mathrm{Arctan}v=\mathrm{Arctan}\left(\frac{u\pm v}{1\mp uv}\right),$$

ⓘ

Symbols:

$\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.34

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

ⓘ

Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

ⓘ

Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

…

14: 10.34 Analytic Continuation

… ►

10.34.3 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=(\mathrm{i}/\pi )\left(\pm {\mathrm{e}}^{m\nu \pi \mathrm{i}}{K}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {\mathrm{e}}^{(m\mp 1)\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i), §10.40(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

►

10.34.4 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

… ►

10.34.6 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\pm {(-1)}^{n(m-1)}m{K}_n\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {(-1)}^{nm}(m\mp 1)K_n\left(z\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

…

15: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. … ►where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. …

16: 3.5 Quadrature

… ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then … ►

Table 3.5.1: Nodes and weights for the 5-point Gauss–Legendre formula.

► ►►

$\pm x_k$		$w_k$
…

►

►

Table 3.5.2: Nodes and weights for the 10-point Gauss–Legendre formula.

► ►►

$\pm x_k$	$w_k$
…

►

►

Table 3.5.3: Nodes and weights for the 20-point Gauss–Legendre formula.

► ►►

$\pm x_k$	$w_k$
…

►

►

Table 3.5.4: Nodes and weights for the 40-point Gauss–Legendre formula.

► ►►

$\pm x_k$	$w_k$
…

►

…

17: 10.36 Other Differential Equations

… ►

10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

…

18: 10.43 Integrals

… ►

$$\int {\mathrm{e}}^{\pm z}{z}^{\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{\nu +1}}{2\nu +1}\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu +1}\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$,

►

$$\int {\mathrm{e}}^{\pm z}{z}^{-\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{-\nu +1}}{1-2\nu }\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu -1}\left(z\right)\right),$$ $\nu \ne \frac{1}{2}$.

►

§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

… ►

§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

… ► …

19: 4.4 Special Values and Limits

… ►

4.4.2 $$\ln \left(-1\pm \mathrm{i}0\right)=\pm \pi \mathrm{i},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.14 (modified)

Permalink:

http://dlmf.nist.gov/4.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

►

4.4.3 $$\ln \left(\pm \mathrm{i}\right)=\pm \frac{1}{2}\pi \mathrm{i}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.15

Permalink:

http://dlmf.nist.gov/4.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

… ►

4.4.6 $${\mathrm{e}}^{\pm \pi \mathrm{i}/2}=\pm \mathrm{i},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

A&S Ref:

4.2.27

Permalink:

http://dlmf.nist.gov/4.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

… ►

4.4.8 $${\mathrm{e}}^{\pm \pi \mathrm{i}/3}=\frac{1}{2}\pm \mathrm{i}\frac{\sqrt{3}}{2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

… ►

4.4.12 $${\mathrm{i}}^{\pm \mathrm{i}}={\mathrm{e}}^{\mp \pi /2}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

…

20: 33.2 Definitions and Basic Properties

… ►

§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

… ►

§33.2(iii) Irregular Solutions $G_{\mathrm{\ell }}(\eta ,\rho ),H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$

►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . Also, ${\mathrm{e}}^{\mp \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{H}_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are analytic functions of $\eta$ when . …