mean value property

♦ 1—10 of 13 matching pages ♦

(0.002 seconds)

1—10 of 13 matching pages

1: 1.9 Calculus of a Complex Variable

… ►

Mean Value Property

…

2: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

►Except for

\nu\left(n\right)

\Lambda\left(n\right)

p_{n}

, and

\pi\left(x\right)

, the functions in §27.2 are multiplicative, which means

f(1)=1

and … ►If

f

is multiplicative, then the values

f(n)

for

n>1

are determined by the values at the prime powers. …Related multiplicative properties are …

3: 4.1 Special Notation

… ► ►►

$k, m, n$	integers.
…

►It is assumed the user is familiar with the definitions and properties of elementary functions of real arguments

x

. The main purpose of the present chapter is to extend these definitions and properties to complex arguments

z

. … ►Sometimes in the literature the meanings of

\ln

and

\operatorname{Ln}

are interchanged; similarly for

\operatorname{arcsin}z

and

\operatorname{Arcsin}z

, etc. …

4: 28.2 Definitions and Basic Properties

§28.2 Definitions and Basic Properties

… ►Other properties are as follows. … ►

Change of Sign of $q$

… ►Period

\pi

means that the eigenfunction has the property

w(z+\pi)=w(z)

, whereas antiperiod

\pi

means that

w(z+\pi)=-w(z)

. Even parity means

w(-z)=w(z)

, and odd parity means

w(-z)=-w(z)

. …

5: 10.41 Asymptotic Expansions for Large Order

… ►As

\nu\to\infty

through positive real values, …where the branches assume their principal values. … ►

§10.41(iv) Double Asymptotic Properties

… ►Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with

z

replaced by

\nu z

, up to and including the term in

z^{-(\ell-1)}

. … ►

§10.41(v) Double Asymptotic Properties (Continued)

…

6: 3.11 Approximation Techniques

… ►They enjoy an orthogonal property with respect to integrals: …as well as an orthogonal property with respect to sums, as follows. … ►For these and further properties of Chebyshev polynomials, see Chapter 18, Gil et al. (2007a, Chapter 3), and Mason and Handscomb (2003). … ►Here the single prime on the summation symbol means that the first term is to be halved. … ►The property …

7: 35.4 Partitions and Zonal Polynomials

… ►For any partition

\kappa

, the zonal polynomial

Z_{\kappa}:\boldsymbol{\mathcal{S}}\to\mathbb{R}

is defined by the properties … ►See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure

\mathrm{d}{\mathbf{H}}

. … ►

§35.4(ii) Properties

… ►

Mean-Value

…

8: 2.1 Definitions and Elementary Properties

§2.1 Definitions and Elementary Properties

… ►(In other words

=

here really means

\subseteq

.) … ►The asymptotic property may also hold uniformly with respect to parameters. … ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. … ►Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …

9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►where the infinite sum means convergence in norm, … ►where the limit has to be understood in the sense of

L^{2}

convergence in the mean: … ►Ignoring the boundary value terms it follows that … ►Boundary values and boundary conditions for the end point

b

are defined in a similar way. If

n_{1}=1

then there are no nonzero boundary values at

a

; if

n_{1}=2

then the above boundary values at

a

form a two-dimensional class. …

10: 1.2 Elementary Algebra

… ►

§1.2(iv) Means

… ►The geometric mean

G

and harmonic mean

H

n

positive numbers

a_{1},a_{2},\dots,a_{n}

are given by … ►If

r

is a nonzero real number, then the weighted mean

M(r)

n

nonnegative numbers

a_{1},a_{2},\dots,a_{n}