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21—30 of 760 matching pages

21: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form … ►Let

T

denote the set of points on

C

that are of finite order (that is, those points

P

for which there exists a positive integer

n

with

nP=o

), and let

I, K

be the sets of points with integer and rational coordinates, respectively. …Both

T, K

are subgroups of

C

, though

I

may not be. …To determine

T

, we make use of the fact that if

(x,y)\in T

then

y^{2}

must be a divisor of

\Delta

; hence there are only a finite number of possibilities for

y

. …The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from

2P=-P

4P=-P

4P=-2P

8P=P

8P=-P

8P=-2P

, or

8P=-4P

. …

22: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function

p\left(n\right)

for

n<N

. … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function

\tau\left(n\right)

, and the values can be checked by the congruence (27.14.20). …

23: 3.8 Nonlinear Equations

… ►The equation to be solved is … ►In this way polynomials of successively lower degree can be used to find the remaining zeros. … ►Newton’s rule can also be used for complex zeros of

p(z)

. However, when the coefficients are all real, complex arithmetic can be avoided by the following iterative process. … ►for solving fixed-point problems (3.8.2) cannot always be predicted, especially in the complex plane. …

24: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►By repeated differentiation of (3.7.1) all derivatives of

w(z)

can be expressed in terms of

w(z)

and

w^{\prime}(z)

as follows. … ►Let

\mathbf{A}_{P}

be the

(2P)\times(2P+2)

band matrix … ►(

\mathbf{I}

and

\boldsymbol{{0}}

being the identity and zero matrices of order

2\times 2

.) … ►If

q(x)

C^{\infty}

on the closure of

(a,b)

, then the discretized form (3.7.13) of the differential equation can be used. …

25: Preface

… ►Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The authoritative status of the existing Handbook, and its orientation toward applications in science, statistics, engineering and computation, will be preserved. ►Thus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. The term digital library has gained acceptance for this kind of information resource, and our choice of project title reflects our hope that the NIST DLMF will be a vehicle for revolutionizing the way applicable mathematics in general is practiced and delivered.

26: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

►

D(m,n)

is the number of paths from

(0,0)

(m,n)

that are composed of directed line segments of the form

(1,0)

(0,1)

, or

(1,1)

. … ►

26.6.12

C\left(n\right)=\sum_{k=1}^{n}N(n,k),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $k$ : nonnegative integer, $n$ : nonnegative integer and $N(n,k)$ : Narayana number
Permalink:: http://dlmf.nist.gov/26.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.13

M(n)=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}C\left(n+1-k\right),

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $n$ : nonnegative integer and $M(n)$ : Motzkin number
Permalink:: http://dlmf.nist.gov/26.6.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.14

C\left(n\right)=\sum_{k=0}^{2n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{2n}{k}M(2n-k).

ⓘ

Symbols:: $C\left(\NVar{n}\right)$ : Catalan number, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $n$ : nonnegative integer and $M(n)$ : Motzkin number
Permalink:: http://dlmf.nist.gov/26.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(iv), §26.6 and Ch.26

27: 3.4 Differentiation

… ►If

f

can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …where

C

is a simple closed contour described in the positive rotational sense such that

C

and its interior lie in the domain of analyticity of

f

, and

x_{0}

is interior to

C

. Taking

C

to be a circle of radius

r

centered at

x_{0}

, we obtain …The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). … ►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. …

28: 28.16 Asymptotic Expansions for Large $q$

… ►Let

s=2m+1

m=0,1,2,\dots

, and

\nu

be fixed with

m<\nu<m+1

. … ►

28.16.1

\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $h$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.16 and Ch.28

…

29: 1.10 Functions of a Complex Variable

… ►Let

f_{1}(z)

be analytic in a domain

D_{1}

. … ►Let

C

be a simple closed contour consisting of a segment

\mathit{AB}

of the real axis and a contour in the upper half-plane joining the ends of

\mathit{AB}

. Also, let

f(z)

be analytic within

C

, continuous within and on

C

, and real on

\mathit{AB}

. … ►Let

F(z)

be a multivalued function and

D

be a domain. … ►Let

D

be a domain and

[a,b]

be a closed finite segment of the real axis. …

30: 26.16 Multiset Permutations

… ►Let

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

be the multiset that has

a_{j}

copies of

j

1\leq j\leq n

. …Additional information can be found in Andrews (1976, pp. 39–45). ►The definitions of inversion number and major index can be extended to permutations of a multiset such as

351322453154\in\mathfrak{S}_{\{1^{2},2^{2},3^{3},4^{2},5^{3}\}}

. Thus

\mathop{\mathrm{inv}}(351322453154)=4+8+0+3+1+1+2+3+1+0+1=24

, and

\mathop{\mathrm{maj}}(351322453154)=2+4+8+9+11=34.

…

21—30 of 760 matching pages

21: 23.20 Mathematical Applications

22: 27.20 Methods of Computation: Other Number-Theoretic Functions

23: 3.8 Nonlinear Equations

24: 3.7 Ordinary Differential Equations

25: Preface

26: 26.6 Other Lattice Path Numbers

Delannoy Number D ⁡ ( m , n )

27: 3.4 Differentiation

28: 28.16 Asymptotic Expansions for Large q

29: 1.10 Functions of a Complex Variable

30: 26.16 Multiset Permutations

Delannoy Number $D(m,n)$

28: 28.16 Asymptotic Expansions for Large $q$