# mean value property

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##### 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 $z^{-(\ell-1)}$. …
##### 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 propertiesSee Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure $\mathrm{d}{\mathbf{H}}$. …
##### 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 $H$ of $n$ positive numbers $a_{1},a_{2},\dots,a_{n}$ are given by … If $r$ is a nonzero real number, then the weighted mean $M(r)$ of $n$ nonnegative numbers $a_{1},a_{2},\dots,a_{n}$, and $n$ positive numbers $p_{1},p_{2},\dots,p_{n}$ with … The scalar product has properties