values%20at%20infinity

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11—12 of 12 matching pages

11: 25.11 Hurwitz Zeta Function

… ►

\zeta\left(s,a\right)

has a meromorphic continuation in the

s

-plane, its only singularity in

\mathbb{C}

being a simple pole at

s=1

with residue

1

. … ►

§25.11(v) Special Values

… ►As

\beta\to\pm\infty

with

s

fixed,

\Re s>1

, …uniformly with respect to bounded nonnegative values of

\alpha

. ►As

a\to\infty

in the sector

|\operatorname{ph}a|\leq\pi-\delta(<\pi)

, with

s(\neq 1)

and

\delta

fixed, we have the asymptotic expansion …

12: Errata

… ►

Section 11.11

The asymptotic results were originally for $\nu$ real valued and $\nu\to+\infty$ . However, these results are also valid for complex values of $\nu$ . The maximum sectors of validity are now specified.

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Subsection 14.2(iii)

Previously the exponents of the associated Legendre differential equation (14.2.2) at infinity were given incorrectly by $\left\{-\nu-1,\nu\right\}$ . These were replaced by $\left\{\nu+1,-\nu\right\}$ .

Reported by Hans Volkmer on 2019-01-30

… ►

Subsection 13.8(iii)

A new paragraph with several new equations and a new reference has been added at the end to provide asymptotic expansions for Kummer functions $U\left(a,b,z\right)$ and ${\mathbf{M}}\left(a,b,z\right)$ as $a\to\infty$ in $|\operatorname{ph}a|\leq\pi-\delta$ and $b$ and $z$ fixed.

… ►

Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

… ►

References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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