removable singularity

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1: 1.10 Functions of a Complex Variable

… ►This singularity is removable if $a_n=0$ for all , and in this case the Laurent series becomes the Taylor series. …Lastly, if $a_n\ne 0$ for infinitely many negative $n$, then $z_0$ is an isolated essential singularity. … ►An isolated singularity $z_0$ is always removable when ${lim}_{z\to z_0}f(z)$ exists, for example $(\sin z)/z$ at $z=0$. …

2: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta =0$, and the Maclaurin series expansion of $c_k(\eta )$ is given by … ►A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by …

3: 1.4 Calculus of One Variable

… ►A removable singularity of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. … …

4: 14.3 Definitions and Hypergeometric Representations

… ►From (15.9.15) it follows that $1-2\mu =0,-1,-2,\mathrm{\dots }$ and $\nu +\mu +1=0,-1,-2,\mathrm{\dots }$ are removable singularities of the right-hand sides of (14.3.21) and (14.3.22).

5: 21.7 Riemann Surfaces

… ►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way. …

6: Errata

… ►

A sentence was added at the end of this subsection indicating that from (15.9.15), it follows that $1-2\mu =0,-1,-2,\mathrm{\dots }$ and $\nu +\mu +1=0,-1,-2,\mathrm{\dots }$ are removable singularities.

…

7: 25.2 Definition and Expansions

… ►It is a meromorphic function whose only singularity in $ℂ$ is a simple pole at $s=1$, with residue 1. … ►

25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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