B%C3%A9zier%20curves

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1: 1.12 Continued Fractions

… ►

A_{n}

and

B_{n}

are called the

n

th (canonical) numerator and denominator respectively. … ►

B_{k}=b_{k}B_{k-1}+a_{k}B_{k-2}

k=1,2,3,\dots

… ►

B_{-1}=0,

►

B_{0}=1.

… ►

b_{1}=B_{1},

…

2: 24.2 Definitions and Generating Functions

… ►

B_{2n+1}=0

… ►

24.2.4

B_{n}=B_{n}\left(0\right),

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.2
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/24.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

►

24.2.5

B_{n}\left(x\right)=\sum_{k=0}^{n}{n\choose k}B_{k}x^{n-k}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.2(i), §24.2 and Ch.24

… ►

\widetilde{B}_{n}\left(x\right)=B_{n}\left(x\right)

… ►

\widetilde{B}_{n}\left(x+1\right)=\widetilde{B}_{n}\left(x\right),

…

3: 3.4 Differentiation

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: … ►

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

… ►

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

►

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

… ►

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

…

4: 26.15 Permutations: Matrix Notation

… ►Let

r_{j}(B)

be the number of ways of placing

j

nonattacking rooks on the squares of

B

. …The rook polynomial is the generating function for

r_{j}(B)

: … ►If

B=B_{1}\cup B_{2}

, where no element of

B_{1}

is in the same row or column as any element of

B_{2}

, then … ►

N(x,B)

is the generating function: …The number of permutations that avoid

B

is …

5: 8.17 Incomplete Beta Functions

… ►

8.17.1

\mathrm{B}_{x}\left(a,b\right)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,\mathrm{d}t,

ⓘ

Defines:: $\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $a$ : parameter, $b$ : parameter and $x$ : variable
A&S Ref:: 6.6.1
Permalink:: http://dlmf.nist.gov/8.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §8.17(i), §8.17 and Ch.8

►

8.17.2

I_{x}\left(a,b\right)=\mathrm{B}_{x}\left(a,b\right)/\mathrm{B}\left(a,b\right),

ⓘ

Defines:: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function
Symbols:: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter and $x$ : variable
A&S Ref:: 6.6.2
Permalink:: http://dlmf.nist.gov/8.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §8.17(i), §8.17 and Ch.8

►where, as in §5.12,

\mathrm{B}\left(a,b\right)

denotes the beta function: … ►For a historical profile of

\mathrm{B}_{x}\left(a,b\right)

see Dutka (1981). … ►

8.17.20

I_{x}\left(a,b\right)=I_{x}\left(a+1,b\right)+\frac{x^{a}(x^{\prime})^{b}}{a% \mathrm{B}\left(a,b\right)},

ⓘ

Symbols:: $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : incomplete beta function, $a$ : parameter, $b$ : parameter, $x$ : variable and $x^{\prime}$
Permalink:: http://dlmf.nist.gov/8.17.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §8.17(iv), §8.17 and Ch.8

…

6: 18.42 Software

… ►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …

7: 26.12 Plane Partitions

… ►We define the

r\times s\times t

box

B(r,s,t)

as …Then the number of plane partitions in

B(r,s,t)

is … ►The number of symmetric plane partitions in

B(r,r,t)

is … ►The number of cyclically symmetric plane partitions in

B(r,r,r)

is … ►The number of descending plane partitions in

B(r,r,r)

is …

8: 25.11 Hurwitz Zeta Function

… ►

25.11.6

\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\,\mathrm{d}x,

s\neq 1

\Re s>-1

a>0

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Keywords:: Euler–Maclaurin formula, improper integral
Source:: Apostol (1985a, (25), p. 231)
Referenced by:: Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)
Permalink:: http://dlmf.nist.gov/25.11.E6
Encodings:: pMML, png, TeX
Errata (effective with 1.0.12):: Originally the integrand was incorrect because its numerator contained the function $\widetilde{B}_{2}\left(x\right)$ . The correct function is $\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{2}$ .
Reported 2016-05-08 by Clemens Heuberger
See also:: Annotations for §25.11(iii), §25.11 and Ch.25

… ►For

\widetilde{B}_{n}\left(x\right)

see §24.2(iii). … ►

25.11.14

\zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right)}{n+1},

n=0,1,2,\dots

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $n$ : nonnegative integer and $a$ : real or complex parameter
Keywords:: special value
Source:: Apostol (1976, (17), p. 264)
Permalink:: http://dlmf.nist.gov/25.11.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(v), §25.11 and Ch.25

… ►

25.11.19

\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}% \right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\,% \mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x% \right)-B_{2}}{(x+a)^{s+2}}\,\mathrm{d}x,

\Re s>-1

s\neq 1

a>0

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, $x$ : real variable, $a$ : real or complex parameter, $s$ : complex variable and $k$ : integer
Keywords:: derivative
Source:: Apostol (1985a, fourth equation, p. 231); with $k=1$ , $\varphi_{2}(x)=\frac{1}{2}(\widetilde{B}_{2}\left(x\right)-B_{2})$
Referenced by:: Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)
Permalink:: http://dlmf.nist.gov/25.11.E19
Encodings:: pMML, png, TeX
Errata (effective with 1.0.12):: Originally both integrands were incorrect because their numerators contained the function $\widetilde{B}_{2}\left(x\right)$ . The correct function is $\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{2}$ .
Reported 2016-06-27 by Gergő Nemes
See also:: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

… ►where

H_{n}

are the harmonic numbers: …

9: 10.20 Uniform Asymptotic Expansions for Large Order

… ►In the following formulas for the coefficients

A_{k}(\zeta)

B_{k}(\zeta)

C_{k}(\zeta)

, and

D_{k}(\zeta)

u_{k}

v_{k}

are the constants defined in §9.7(i), and

U_{k}(p)

V_{k}(p)

are the polynomials in

p

of degree

3k

defined in §10.41(ii). … ►Note: Another way of arranging the above formulas for the coefficients

A_{k}(\zeta),B_{k}(\zeta),C_{k}(\zeta)

, and

D_{k}(\zeta)

would be by analogy with (12.10.42) and (12.10.46). … ►Each of the coefficients

A_{k}(\zeta)

B_{k}(\zeta)

C_{k}(\zeta)

, and

D_{k}(\zeta)

k=0,1,2,\dotsc

, is real and infinitely differentiable on the interval

-\infty<\zeta<\infty

. … ►The curves

BP_{1}E_{1}

and

BP_{2}E_{2}

in the

z

-plane are the inverse maps of the line segments … ►The eye-shaped closed domain in the uncut

z

-plane that is bounded by

BP_{1}E_{1}

and

BP_{2}E_{2}

is denoted by

\mathbf{K}

; see Figure 10.20.3. …

10: 24.4 Basic Properties

… ►

24.4.1

B_{n}\left(x+1\right)-B_{n}\left(x\right)=nx^{n-1},

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.6
Referenced by:: (13.8.16), §24.13(i)
Permalink:: http://dlmf.nist.gov/24.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(i), §24.4 and Ch.24

… ►

24.4.3

B_{n}\left(1-x\right)=(-1)^{n}B_{n}\left(x\right),

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.8
Permalink:: http://dlmf.nist.gov/24.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(ii), §24.4 and Ch.24

… ►

24.4.25

B_{n}\left(0\right)=(-1)^{n}B_{n}\left(1\right)=B_{n}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer
A&S Ref:: 23.1.20
Referenced by:: §24.17(ii), §24.4(vi)
Permalink:: http://dlmf.nist.gov/24.4.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(vi), §24.4 and Ch.24

… ►Let

P(x)

denote any polynomial in

x

, and after expanding set

(B(x))^{n}=B_{n}\left(x\right)

and

(E(x))^{n}=E_{n}\left(x\right)

. … ►

24.4.37

B_{n}\left(x+h\right)=(B(x)+h)^{n},

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.26
Permalink:: http://dlmf.nist.gov/24.4.E37
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(viii), §24.4 and Ch.24

…