# singular

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## 21—30 of 82 matching pages

##### 21: 31.6 Path-Multiplicative Solutions
This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the $z$-plane that encircles $s_{1}$ and $s_{2}$ once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor ${\mathrm{e}}^{2\nu\pi i}$. …
##### 22: 14.8 Behavior at Singularities
###### §14.8 Behavior at Singularities
14.8.16 ${\boldsymbol{Q}^{\mu}_{-n-(1/2)}\left(x\right)\sim\frac{\pi^{1/2}\Gamma\left(% \mu+n+\frac{1}{2}\right)}{n!\Gamma\left(\mu-n+\frac{1}{2}\right)(2x)^{n+(1/2)}% }},$ $n=1,2,3,\dots$, $\mu-n+\frac{1}{2}\neq 0,-1,-2,\dots$.
##### 25: 29.2 Differential Equations
This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K^{\prime}}$, where $p,q\in\mathbb{Z}$, and $K$, ${K^{\prime}}$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime}(=(1-k^{2})^{1/2})$, respectively; see §19.2(ii). In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). …
##### 26: 31.2 Differential Equations
31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
This equation has regular singularities at $0,1,a,\infty$, with corresponding exponents $\{0,1-\gamma\}$, $\{0,1-\delta\}$, $\{0,1-\epsilon\}$, $\{\alpha,\beta\}$, respectively (§2.7(i)). All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $\mathbb{C}\cup\{\infty\}$, can be transformed into (31.2.1). The parameters play different roles: $a$ is the singularity parameter; $\alpha,\beta,\gamma,\delta,\epsilon$ are exponent parameters; $q$ is the accessory parameter. …
##### 27: 10.25 Definitions
This equation is obtained from Bessel’s equation (10.2.1) on replacing $z$ by $\pm iz$, and it has the same kinds of singularities. … …
##### 28: 10.2 Definitions
###### §10.2(i) Bessel’s Equation
This differential equation has a regular singularity at $z=0$ with indices $\pm\nu$, and an irregular singularity at $z=\infty$ of rank $1$; compare §§2.7(i) and 2.7(ii). …
##### 29: 31.3 Basic Solutions
In general, one of them has a logarithmic singularity at $z=0$.
###### §31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
Solutions (31.3.1) and (31.3.5)–(31.3.11) comprise a set of 8 local solutions of (31.2.1): 2 per singular point. …
##### 30: 32.2 Differential Equations
be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $\ifrac{\mathrm{d}w}{\mathrm{d}z}$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $\mathbb{C}$. In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities1.10(iii)), however. …