# behavior at singularities

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## 1—10 of 11 matching pages

##### 1: 14.8 Behavior at Singularities
###### §14.8 BehavioratSingularities
14.8.16 ${\boldsymbol{Q}^{\mu}_{-n-(1/2)}\left(x\right)\sim\frac{\pi^{1/2}\Gamma\left(% \mu+n+\frac{1}{2}\right)}{n!\Gamma\left(\mu-n+\frac{1}{2}\right)(2x)^{n+(1/2)}% }},$ $n=1,2,3,\dots$, $\mu-n+\frac{1}{2}\neq 0,-1,-2,\dots$.
##### 2: 14.21 Definitions and Basic Properties
###### §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). …
##### 4: 19.12 Asymptotic Approximations
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: …
##### 5: 2.10 Sums and Sequences
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). … We seek the behavior as $x\to+\infty$. …
• (b´)

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.

• The singularities of $f(z)$ on the unit circle are branch points at $z=e^{\pm i\alpha}$. To match the limiting behavior of $f(z)$ at these points we set …
##### 6: 8.12 Uniform Asymptotic Expansions for Large Parameter
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 … For the asymptotic behavior of $c_{k}(\eta)$ as $k\to\infty$ see Dunster et al. (1998) and Olde Daalhuis (1998c). … A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by …
##### 7: 2.7 Differential Equations
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … 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 $t=\infty$, 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
###### 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 $f(x-)\neq f(x+)$; see §6.16(i). …
##### 9: Bibliography B
• M. V. Berry (1981) 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.
• N. Bleistein (1966) Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, pp. 353–370.
• N. Bleistein (1967) Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities. J. Math. Mech. 17, pp. 533–559.
• R. Bo and R. Wong (1996) Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.
• W. Bühring (1987b) The behavior at unit argument of the hypergeometric function ${}_{3}F_{2}$ . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.
• ##### 10: 2.3 Integrals of a Real Variable
Other types of singular behavior in the integrand can be treated in an analogous manner. … Without loss of generality, we assume that this minimum is at the left endpoint $a$. … 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. … For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). … it is free from singularity at $t=\alpha$. …