# mean value property for harmonic functions

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###### §19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
As $n\to\infty$, $a_{n}$ and $g_{n}$ converge to a common limit $M\left(a_{0},g_{0}\right)$ called the AGM (Arithmetic-Geometric Mean) of $a_{0}$ and $g_{0}$. …showing that the convergence of $c_{n}$ to 0 and of $a_{n}$ and $g_{n}$ to $M\left(a_{0},g_{0}\right)$ is quadratic in each case. … Again, $p_{n}$ and $\varepsilon_{n}$ converge quadratically to $M\left(a_{0},g_{0}\right)$ and 0, respectively, and $Q_{n}$ converges to 0 faster than quadratically. If $\alpha^{2}>1$, then the Cauchy principal value is …
##### 2: 14.30 Spherical and Spheroidal Harmonics
###### §14.30(ii) Basic Properties
Most mathematical properties of $Y_{{l},{m}}\left(\theta,\phi\right)$ can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. …
##### 3: 9.12 Scorer Functions
###### §9.12(iii) Initial Values
where the last integral is a Cauchy principal value1.4(v)). … For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c). …
##### 4: 5.2 Definitions
###### Euler’s Integral
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\,\mathrm{d}t,$ $\Re z>0$.
It is a meromorphic function with no zeros, and with simple poles of residue $(-1)^{n}/n!$ at $z=-n$. …
5.2.3 $\gamma=\lim_{n\to\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}-\ln n% \right)=0.57721\;56649\;01532\;86060\;\dots.$
##### 5: 5.15 Polygamma Functions
###### §5.15 Polygamma Functions
The functions $\psi^{(n)}\left(z\right)$, $n=1,2,\dots$, are called the polygamma functions. In particular, $\psi'\left(z\right)$ is the trigamma function; $\psi''$, $\psi^{(3)}$, $\psi^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … For $B_{2k}$ see §24.2(i). …
##### 6: 15.2 Definitions and Analytical Properties
###### §15.2(i) Gauss Series
For all values of $c$
###### §15.2(ii) Analytic Properties
Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …
##### 7: 11.9 Lommel Functions
###### §11.9 Lommel Functions
Another solution of (11.9.1) that is defined for all values of $\mu$ and $\nu$ is $S_{{\mu},{\nu}}\left(z\right)$, where … … For descriptive properties of $s_{{\mu},{\nu}}\left(x\right)$ see Steinig (1972). …
##### 8: 23.15 Definitions
###### §23.15 Definitions
A modular function $f(\tau)$ is a function of $\tau$ that is meromorphic in the half-plane $\Im\tau>0$, and has the property that for all $\mathcal{A}\in\mbox{SL}(2,\mathbb{Z})$, or for all $\mathcal{A}$ belonging to a subgroup of SL$(2,\mathbb{Z})$, … …
##### 9: 5.12 Beta Function
###### §5.12 Beta Function
In this section all fractional powers have their principal values, except where noted otherwise. … In (5.12.8) the fractional powers have their principal values when $w>0$ and $z>0$, and are continued via continuity. … In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning. …
##### 10: 20.2 Definitions and Periodic Properties
###### §20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …