# singularity

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## 11—20 of 82 matching pages

##### 11: 30.2 Differential Equations
This equation has regular singularities at $z=\pm 1$ with exponents $\pm\frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\infty$ (if $\gamma\neq 0$). … …
##### 12: Mark J. Ablowitz
ODEs which do not have moveable branch point singularities. …
##### 13: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
###### §31.5 Solutions Analytic at Three Singularities: Heun Polynomials
is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. …
##### 14: 31.15 Stieltjes Polynomials
31.15.2 $\sum_{j=1}^{N}\frac{\gamma_{j}/2}{z_{k}-a_{j}}+\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{n}\frac{1}{z_{k}-z_{j}}=0,$ $k=1,2,\dots,n$.
31.15.3 $\sum_{j=1}^{N}\frac{\gamma_{j}}{t_{k}-a_{j}}+\sum_{j=1}^{n-1}\frac{1}{t_{k}-z_% {j}^{\prime}}=0.$
31.15.6 $a_{j} $j=1,2,\dots,N-1$,
31.15.7 $q_{j}=\gamma_{j}\sum_{k=1}^{n}\frac{1}{z_{k}-a_{j}},$ $j=1,2,\dots,N$.
If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})$, where each $m_{j}$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_{j}$ zeros in the open interval $(a_{j},a_{j+1})$ for each $j=1,2,\dots,N-1$. …
##### 15: 2.7 Differential Equations
###### §2.7(i) Regular Singularities: Fuchs–Frobenius Theory
Other points $z_{0}$ are singularities of the differential equation. …All other singularities are classified as irregular. …
###### §2.7(ii) Irregular Singularities of Rank 1
Thus a regular singularity has rank 0. …
##### 16: 15.11 Riemann’s Differential Equation
###### §15.11(i) Equations with Three Singularities
The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. … The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …
##### 17: 10.72 Mathematical Applications
These expansions are uniform with respect to $z$, including the turning point $z_{0}$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of $z_{0}$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. …
##### 18: Bibliography O
• A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
• A. B. Olde Daalhuis (1998a) Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
• F. W. J. Olver and F. Stenger (1965) Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.
• F. W. J. Olver (1965) On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
• F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.
• ##### 19: 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). …
##### 20: 15.17 Mathematical Applications
The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …