mean value property

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1: 1.9 Calculus of a Complex Variable

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Mean Value Property

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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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$k,m,n$	integers.
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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 $z$. … ►Sometimes in the literature the meanings of $\ln$ and $\mathrm{Ln}$ are interchanged; similarly for $\arcsin z$ and $\mathrm{Arcsin}z$, 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 $\nu \to \mathrm{\infty }$ 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 $z^{-(\mathrm{\ell }-1)}$. … ►

§10.41(v) Double Asymptotic Properties (Continued)

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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 $Z_{\kappa }:𝓢\to ℝ$ is defined by the properties … ►See Muirhead (1982, pp. 68–72) for the definition and properties of the Haar measure $\mathrm{d}𝐇$. … ►

§35.4(ii) Properties

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Mean-Value

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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: 1.2 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 … ►

Special Properties and Definitions Relating to Square Matrices

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10: 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. …