# mean value property

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##### 1: 1.9 Calculus of a Complex Variable

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###### Mean Value Property

…##### 2: 27.3 Multiplicative Properties

###### §27.3 Multiplicative Properties

►Except for $\nu \left(n\right)$, $\mathrm{\Lambda}\left(n\right)$, ${p}_{n}$, 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

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►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 $z$.
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►Sometimes in the literature the meanings of $\mathrm{ln}$ and $\mathrm{Ln}$ are interchanged; similarly for $\mathrm{arcsin}z$ and $\mathrm{Arcsin}z$, etc.
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$k,m,n$ | integers. |
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##### 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

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►As $\nu \to \mathrm{\infty}$ through positive real values,
…where the branches assume their principal values.
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###### §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 ${z}^{-(\mathrm{\ell}-1)}$. … ►###### §10.41(v) Double Asymptotic Properties (Continued)

…##### 6: 3.11 Approximation Techniques

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►They enjoy an orthogonal property with respect to integrals:
…as well as an orthogonal property with respect to sums, as follows.
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►For these and further properties of Chebyshev polynomials, see Chapter 18, Gil et al. (2007a, Chapter 3), and Mason and Handscomb (2003).
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►Here the single prime on the summation symbol means that the first term is to be halved.
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►The property
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##### 7: 35.4 Partitions and Zonal Polynomials

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►For any partition $\kappa $, the

*zonal polynomial*${Z}_{\kappa}:\U0001d4e2\to \mathbb{R}$ 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

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►where the infinite sum means convergence in norm,
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►where the limit has to be understood in the sense of ${L}^{2}$ convergence in the mean:
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►Ignoring the boundary value terms it follows that
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►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.
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##### 10: 1.2 Elementary Algebra

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###### §1.2(iv) Means

… ►The*geometric mean*$G$ and*harmonic mean*$H$ of $n$ positive numbers ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$ are given by … ►If $r$ is a nonzero real number, then the*weighted mean*$M(r)$ of $n$ nonnegative numbers ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$, and $n$ positive numbers ${p}_{1},{p}_{2},\mathrm{\dots},{p}_{n}$ with … ►The scalar product has properties … ►