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1: 1.10 Functions of a Complex Variable

… ►Let $r_1=0$, so that the annulus becomes the punctured neighborhood $N$: , and assume that $f(z)$ is analytic in $N$, but not at $z_0$. …Lastly, if $a_n\ne 0$ for infinitely many negative $n$, then $z_0$ is an isolated essential singularity. … ►Suppose $F(z)$ is multivalued and $a$ is a point such that there exists a branch of $F(z)$ in a cut neighborhood of $a$, but there does not exist a branch of $F(z)$ in any punctured neighborhood of $a$. …

2: 2.7 Differential Equations

… ►In a punctured neighborhood $𝐍$ of a regular singularity $z_0$ … ►

2.7.4 $$w_j(z)={(z-z_0)}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}{a}_{s,j}{(z-z_0)}^s,$$ $z\in 𝐍$,

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Symbols:

$\in$: element of, $𝐍$: punctured neighborhood, $w_j(z)$: solutions and $a_n$: coefficients

Referenced by:

§2.7(i), §2.7(i)

Permalink:

http://dlmf.nist.gov/2.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(i), §2.7 and Ch.2

… ►

2.7.6 $$w_2(z)={(z-z_0)}^{{\alpha }_2}\sum _{\begin{array}{c}s=0\\ s\ne {\alpha }_1-{\alpha }_2\end{array}}^{\mathrm{\infty }}{b}_s{(z-z_0)}^s+c{w}_1(z)\ln \left(z-z_0\right),$$ $z\in 𝐍$.

ⓘ

Symbols:

$\in$: element of, $\ln z$: principal branch of logarithm function, $b_s$: coefficients, $c$: constant, $𝐍$: punctured neighborhood and $w_j(z)$: solutions

Referenced by:

§2.7(i), §2.9(ii)

Permalink:

http://dlmf.nist.gov/2.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(i), §2.7 and Ch.2

…