# classification of singularities

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9 matching pages ♦

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## 9 matching pages

##### 1: 16.8 Differential Equations

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###### §16.8(i) Classification of Singularities

…##### 2: 2.7 Differential Equations

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►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.
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##### 3: 1.13 Differential Equations

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►For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7.
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##### 4: 3.7 Ordinary Differential Equations

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►For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7.
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##### 5: 31.2 Differential Equations

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►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $\u2102\cup \{\mathrm{\infty}\}$, can be transformed into (31.2.1).
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##### 6: 16.21 Differential Equation

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►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 $z=\mathrm{\infty}$.
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##### 7: 31.14 General Fuchsian Equation

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►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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##### 8: 31.12 Confluent Forms of Heun’s Equation

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