# classification of singularities

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##### 2: 2.7 Differential Equations
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. …
##### 3: 3.7 Ordinary Differential Equations
For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
##### 4: 1.13 Differential Equations
For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …
##### 5: 31.2 Differential Equations
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). …
##### 6: 16.21 Differential Equation
With the classification of §16.8(i), when $p the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\infty$. …
##### 7: 31.14 General Fuchsian Equation
The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_{j}$, $j=1,2,\dots,N$, and at $\infty$, is given by …The exponents at the finite singularities $a_{j}$ are $\{0,{1-\gamma_{j}}\}$ and those at $\infty$ are $\{\alpha,\beta\}$, where …
$\alpha\beta=\sum_{j=1}^{N}a_{j}q_{j}.$
The three sets of parameters comprise the singularity parameters $a_{j}$, the exponent parameters $\alpha,\beta,\gamma_{j}$, and the $N-2$ free accessory parameters $q_{j}$. …
31.14.3 $w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),$
##### 8: 31.12 Confluent Forms of Heun’s Equation
Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\infty$. … This has irregular singularities at $z=0$ and $\infty$, each of rank $1$. … This has a regular singularity at $z=0$, and an irregular singularity at $\infty$ of rank $2$. … This has one singularity, an irregular singularity of rank $3$ at $z=\infty$. …
##### 9: 2.8 Differential Equations with a Parameter
###### §2.8(i) Classification of Cases
The form of the asymptotic expansion depends on the nature of the transition points in $\mathbf{D}$, that is, points at which $f(z)$ has a zero or singularity. …