classification of singularities

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1: 16.8 Differential Equations

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§16.8(i) 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, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). …

6: 16.21 Differential Equation

… ►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\mathrm{\infty }$. …

7: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by …The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where … ►

$\alpha \beta ={\displaystyle \sum _{j=1}^N}{a}_j{q}_j.$

►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 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

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See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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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 $z=\mathrm{\infty }$. … ►This has irregular singularities at $z=0$ and $\mathrm{\infty }$, each of rank $1$. … ►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty }$ of rank $2$. … ►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty }$. …

9: 2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

… ►The form of the asymptotic expansion depends on the nature of the transition points in $𝐃$, that is, points at which $f(z)$ has a zero or singularity. …