# dominated convergence theorem

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##### 1: 1.9 Calculus of a Complex Variable
###### Cauchy’s Theorem
Absolutely convergent series are also convergent. …
##### 2: 2.7 Differential Equations
For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:
###### Liouville–Green Approximation Theorem
$w_{1}(x)$ is a recessive (or subdominant) solution as $x\to a_{1}+$, and $w_{4}(x)$ is a dominant solution as $x\to a_{1}+$. … as $x\to+\infty$, $w_{2}(x)$ being recessive and $w_{3}(x)$ dominant. … The solutions $w_{1}(z)$ and $w_{2}(z)$ are respectively recessive and dominant as $\Re z\to-\infty$, and vice versa as $\Re z\to+\infty$. …
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
##### 4: 3.8 Nonlinear Equations
The rule converges locally and is cubically convergent. … The convergence of iterative methods …
##### 7: 1.2 Elementary Algebra
###### Binomial Theorem
Square $n\times n$ matrices (said to be of order $n$ ) dominate the use of matrices in the DLMF, and they have many special properties. … which converges, entry-wise or in norm, for all $\mathbf{A}$. …
##### 8: 14.28 Sums
The series converges uniformly for $z_{1}$ outside or on $\mathcal{E}_{1}$, and $z_{2}$ within or on $\mathcal{E}_{2}$. For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …
Then (35.2.1) converges absolutely on the region $\Re\left(\mathbf{Z}\right)>\mathbf{X}_{0}$, and $g(\mathbf{Z})$ is a complex analytic function of all elements $z_{j,k}$ of $\mathbf{Z}$. … Assume that $\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{U}+\mathrm{i}\mathbf{V})\right|% \,\mathrm{d}{\mathbf{V}}$ converges, and also that its limit as $\mathbf{U}\to\infty$ is $0$. …
This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $\mathbb{C}$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $|\operatorname{ph}{z}|<\pi$. …