# M-test for uniform convergence

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##### 1: T. Mark Dunster

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►He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory.
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##### 2: 36.15 Methods of Computation

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###### §36.15(i) Convergent Series

… ►Close to the bifurcation set but far from $\mathbf{x}=\mathrm{\U0001d7ce}$, the uniform asymptotic approximations of §36.12 can be used. …##### 3: 14.32 Methods of Computation

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►In particular, for small or moderate values of the parameters $\mu $ and $\nu $ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real.
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Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

##### 4: 8.25 Methods of Computation

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►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation.
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►DiDonato and Morris (1986) describes an algorithm for computing $P(a,x)$ and $Q(a,x)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ from the uniform expansions in §8.12.
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##### 5: 1.10 Functions of a Complex Variable

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►The convergence of the infinite product is

*uniform*if the sequence of partial products converges uniformly. ►###### $M$-test

… ►where ${a}_{n}(z)$ is analytic for all $n\ge 1$, and the convergence of the product is uniform in any compact subset of $D$. …##### 6: 1.9 Calculus of a Complex Variable

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►Absolutely convergent series are also convergent.
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►The sequence

*converges uniformly*on $S$, if for every $\u03f5>0$ there exists an integer $N$, independent of $z$, such that … ►###### Weierstrass $M$-test

… ►For $z$ in $\left|z-{z}_{0}\right|\le \rho $ ($$), the convergence is absolute and uniform. … ► …##### 7: 14.26 Uniform Asymptotic Expansions

###### §14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\bm{Q}}_{\nu}^{\mu}\left(x\right)$ for $$ are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.##### 8: Bibliography D

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Uniform asymptotic expansions for prolate spheroidal functions with large parameters.
SIAM J. Math. Anal. 17 (6), pp. 1495–1524.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Uniform asymptotic approximation of Mathieu functions.
Methods Appl. Anal. 1 (2), pp. 143–168.
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Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
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Uniform asymptotic approximations for incomplete Riemann zeta functions.
J. Comput. Appl. Math. 190 (1-2), pp. 339–353.
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##### 9: 2.1 Definitions and Elementary Properties

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►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions.
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►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series.
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###### §2.1(iv) Uniform Asymptotic Expansions

… ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. … ►It can even happen that a generalized asymptotic expansion converges, but its sum is not the function being represented asymptotically; for an example see §18.15(iii).##### 10: 33.20 Expansions for Small $|\u03f5|$

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►The series (33.20.3) converges for all $r$ and $\u03f5$.
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