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1: 2.10 Sums and Sequences

… ►Let $f(z)$ be analytic on the annulus , with Laurent expansion ►

2.10.25 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{z}^n,$$ .

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Symbols:

$f(x)$: analytic function, $f_n$: coefficients and $r$: radious of annulus

Permalink:

http://dlmf.nist.gov/2.10.E25

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See also:

Annotations for §2.10(iv), §2.10 and Ch.2

… ►where $𝒞$ is a simple closed contour in the annulus that encloses $z=0$. … ►

The coefficients in the Laurent expansion

2.10.27 $$g(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{g}_n{z}^n,$$ ,

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Symbols:

$r$: radious of annulus, $g(z)$: comparison function and $g_n$: coefficients

Permalink:

http://dlmf.nist.gov/2.10.E27

Encodings:

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Annotations for §2.10(iv), §2.10 and Ch.2

have known asymptotic behavior as $n\to \pm \mathrm{\infty }$.

… ►

2.10.29 $$f_n=g_n+o\left(r^{-n}\right),$$ $n\to \pm \mathrm{\infty }$.

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Symbols:

$o\left(x\right)$: order less than, $f_n$: coefficients, $r$: radious of annulus and $g_n$: coefficients

Referenced by:

§2.10(iv)

Permalink:

http://dlmf.nist.gov/2.10.E29

Encodings:

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Annotations for §2.10(iv), §2.10 and Ch.2

…

2: 1.10 Functions of a Complex Variable

… ►

§1.10(iii) Laurent Series

►Suppose $f(z)$ is analytic in the annulus , , and $r\in (r_1,r_2)$. Then …The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. ►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 $z_0$. …

3: 2.7 Differential Equations

… ►these series converging in an annulus $|z|>a$, with at least one of $f_0$, $g_0$, $g_1$ nonzero. …