# behavior at singularities

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##### 1: 14.8 Behavior at Singularities

###### §14.8 Behavior at Singularities

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14.8.16
$${\mathit{Q}}_{-n-(1/2)}^{\mu}\left(x\right)\sim \frac{{\pi}^{1/2}\mathrm{\Gamma}\left(\mu +n+\frac{1}{2}\right)}{n!\mathrm{\Gamma}\left(\mu -n+\frac{1}{2}\right){(2x)}^{n+(1/2)}},$$
$n=1,2,3,\mathrm{\dots}$, $\mu -n+\frac{1}{2}\ne 0,-1,-2,\mathrm{\dots}$.

##### 2: 14.21 Definitions and Basic Properties

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###### §14.21(iii) Properties

… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …##### 3: 14.20 Conical (or Mehler) Functions

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###### §14.20(iii) Behavior as $x\to 1$

…##### 4: 19.12 Asymptotic Approximations

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►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at
$k=1$ is given by the following convergent series:
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##### 5: 2.10 Sums and Sequences

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►For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at
$x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a).
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►We seek the behavior as $x\to +\mathrm{\infty}$.
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(b´)
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►The singularities of $f(z)$ on the unit circle are branch points at
$z={\mathrm{e}}^{\pm \mathrm{i}\alpha}$.
To match the limiting behavior of $f(z)$
at these points we set
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On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity ${z}_{j}$, say,

2.10.30
$$f(z)-g(z)=O\left({(z-{z}_{j})}^{{\sigma}_{j}-1}\right),$$
$z\to {z}_{j}$,

where ${\sigma}_{j}$ is a positive constant.

##### 6: 8.12 Uniform Asymptotic Expansions for Large Parameter

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►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at
$\eta =0$, and the Maclaurin series expansion of ${c}_{k}(\eta )$ is given by
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►For the asymptotic behavior of ${c}_{k}(\eta )$ as $k\to \mathrm{\infty}$ see Dunster et al. (1998) and Olde Daalhuis (1998c).
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►A different type of uniform expansion with coefficients that do not possess a removable singularity at
$z=a$ is given by
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##### 7: 2.7 Differential Equations

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►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.
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►If both $(z-{z}_{0})f(z)$ and ${(z-{z}_{0})}^{2}g(z)$ are analytic at
${z}_{0}$, then ${z}_{0}$ is a

*regular singularity*(or*singularity of the first kind*). … ►The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … ►The transformed differential equation either has a regular singularity at $t=\mathrm{\infty}$, or its characteristic equation has unequal roots. … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …##### 8: 1.8 Fourier Series

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###### Lebesgue Constants

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda $. … ►Let $f(x)$ be an absolutely integrable function of period $2\pi $, and continuous except at a finite number of points in any bounded interval. …at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). The convergence is non-uniform, however, at points where $f(x-)\ne f(x+)$; see §6.16(i). …##### 9: Bibliography B

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Singularities in Waves and Rays.
In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.),
Vol. 35, pp. 453–543.
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Uniform asymptotic expansions of integrals with stationary point near algebraic singularity.
Comm. Pure Appl. Math. 19, pp. 353–370.
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Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities.
J. Math. Mech. 17, pp. 533–559.
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Asymptotic behavior of the Pollaczek polynomials and their zeros.
Stud. Appl. Math. 96, pp. 307–338.
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The behavior at unit argument of the hypergeometric function ${}_{3}F_{2}$
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SIAM J. Math. Anal. 18 (5), pp. 1227–1234.
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##### 10: 2.3 Integrals of a Real Variable

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►Other types of singular behavior in the integrand can be treated in an analogous manner.
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►Without loss of generality, we assume that this minimum is at the left endpoint $a$.
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►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint.
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►For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010).
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►it is free from singularity at
$t=\alpha $.
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