on general sets

SearchAdvancedHelp

(0.005 seconds)

1—10 of 252 matching pages

1: 18.2 General Orthogonal Polynomials

… ►A system (or set) of polynomials $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, where $p_n(x)$ has degree $n$ as in §18.1(i), is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\ge 0$) if … ►

Orthogonality on Countable Sets

… ►

Orthogonality on General Sets

…

2: 9.13 Generalized Airy Functions

… ►

$$A_n\left(0\right)=p^{1/2}{B}_n\left(0\right)=\frac{p^{1-p}}{\mathrm{\Gamma }\left(1-p\right)},$$

… ►

9.13.15

ⓘ

Symbols:

$A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $z$: complex variable, $n$: parameter, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $m$: index and $V_m$: Olver’s generalized Airy function

Permalink:

http://dlmf.nist.gov/9.13.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.16

ⓘ

Symbols:

$B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $z$: complex variable, $n$: parameter, $U_{\mathrm{i}}$: Smirnov’s generalized Airy function, $m$: index and $V_m$: Olver’s generalized Airy function

Permalink:

http://dlmf.nist.gov/9.13.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►

$A_2(z,p)={\mathrm{e}}^{-2(p-1)\pi \mathrm{i}/3}{A}_1(z{\mathrm{e}}^{2\pi \mathrm{i}/3},p)$ ,

►

$A_3(z,p)={\mathrm{e}}^{2(p-1)\pi \mathrm{i}/3}{A}_1(z{\mathrm{e}}^{-2\pi \mathrm{i}/3},p)$ .

…

3: 36.5 Stokes Sets

… ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►The second sheet corresponds to $x>0$ and it intersects the bifurcation set (§36.4) smoothly along the line generated by $X=X_1=6.95643$, $\left|Y\right|=\left|Y_1\right|=6.81337$. … ►the intersection lines with the bifurcation set are generated by $|X|=X_2=0.45148$, $Y=Y_2=0.59693$. …

4: 3.8 Nonlinear Equations

… ►For an arbitrary starting point $z_0\in ℂ$, convergence cannot be predicted, and the boundary of the set of points $z_0$ that generate a sequence converging to a particular zero has a very complicated structure. …In general the Julia set of an analytic function $f(z)$ is a fractal, that is, a set that is self-similar. …

5: 4.12 Generalized Logarithms and Exponentials

… ►For $C^{\mathrm{\infty }}$ generalized logarithms, see Walker (1991). …

6: Bibliography S

… ►

Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.

ⓘ

External Links:

Document

Referenced by:

§34.13.

Encodings:

BibTeX

…

7: 4.37 Inverse Hyperbolic Functions

… ►

4.37.1 $$\mathrm{Arcsinh}z={\int }_0^z\frac{dt}{{(1+t^2)}^{1/2}},$$

ⓘ

Defines:

$\mathrm{Arcsinh}z$: general inverse hyperbolic sine function

Symbols:

$dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Referenced by:

§4.37(i), §4.37(ii)

Permalink:

http://dlmf.nist.gov/4.37.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

►

4.37.2 $$\mathrm{Arccosh}z={\int }_1^z\frac{dt}{{(t^2-1)}^{1/2}},$$

ⓘ

Defines:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function

Symbols:

$dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Referenced by:

§4.37(i)

Permalink:

http://dlmf.nist.gov/4.37.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

… ►

4.37.4 $$\mathrm{Arccsch}z=\mathrm{Arcsinh}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arccsch}z$: general inverse hyperbolic cosecant function

Symbols:

$\mathrm{Arcsinh}z$: general inverse hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

►

4.37.5 $$\mathrm{Arcsech}z=\mathrm{Arccosh}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arcsech}z$: general inverse hyperbolic secant function

Symbols:

$\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

►

4.37.6 $$\mathrm{Arccoth}z=\mathrm{Arctanh}\left(1/z\right).$$

ⓘ

Defines:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function

Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(i), §4.37 and Ch.4

…

8: 4.8 Identities

… ►

4.8.5 $$\mathrm{Ln}\left(z^n\right)=n\mathrm{Ln}z,$$ $n\in \mathbb{Z}$,

ⓘ

Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $n$: integer and $z$: complex variable

A&S Ref:

4.1.10

Referenced by:

§4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.8 $$\mathrm{Ln}\left(\exp z\right)=z+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer and $z$: complex variable

A&S Ref:

4.2.2

Permalink:

http://dlmf.nist.gov/4.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►

4.8.11 $$\mathrm{Ln}\left(a^z\right)=z\mathrm{Ln}a+2k\pi \mathrm{i},$$ $k\in \mathbb{Z}$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\mathrm{Ln}z$: general logarithm function, $k$: integer, $a$: real or complex constant and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

…

9: 4.23 Inverse Trigonometric Functions

… ►

4.23.1 $$\mathrm{Arcsin}z={\int }_0^z\frac{dt}{{(1-t^2)}^{1/2}},$$

ⓘ

Defines:

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

Symbols:

$dx$: differential of $x$ , $\int$: integral and $z$: complex variable

A&S Ref:

4.4.1 (modified)

Referenced by:

§4.23(i), §4.23(ii), §4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

►

4.23.2 $$\mathrm{Arccos}z={\int }_z^1\frac{dt}{{(1-t^2)}^{1/2}},$$

ⓘ

Defines:

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

Symbols:

$dx$: differential of $x$ , $\int$: integral and $z$: complex variable

A&S Ref:

4.4.2 (modified)

Referenced by:

§4.23(i)

Permalink:

http://dlmf.nist.gov/4.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

… ►

4.23.4 $$\mathrm{Arccsc}z=\mathrm{Arcsin}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arccsc}z$: general arccosecant function

Symbols:

$\mathrm{Arcsin}z$: general arcsine function and $z$: complex variable

A&S Ref:

4.4.6 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

►

4.23.5 $$\mathrm{Arcsec}z=\mathrm{Arccos}\left(1/z\right),$$

ⓘ

Defines:

$\mathrm{Arcsec}z$: general arcsecant function

Symbols:

$\mathrm{Arccos}z$: general arccosine function and $z$: complex variable

A&S Ref:

4.4.7 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

►

4.23.6 $$\mathrm{Arccot}z=\mathrm{Arctan}\left(1/z\right).$$

ⓘ

Defines:

$\mathrm{Arccot}z$: general arccotangent function

Symbols:

$\mathrm{Arctan}z$: general arctangent function and $z$: complex variable

A&S Ref:

4.4.8 (modified)

Permalink:

http://dlmf.nist.gov/4.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(i), §4.23 and Ch.4

…

10: 36.4 Bifurcation Sets

§36.4 Bifurcation Sets

… ►

Bifurcation (Catastrophe) Set for Cuspoids

… ►

Bifurcation (Catastrophe) Set for Umbilics

… ► $K=1$, fold bifurcation set: … ►

§36.4(ii) Visualizations

…