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31: 25.15 Dirichlet $L$-functions

… ►For the principal character ${\chi }_1(modk)$, $L(s,{\chi }_1)$ is analytic everywhere except for a simple pole at $s=1$ with residue $\varphi \left(k\right)/k$, where $\varphi \left(k\right)$ is Euler’s totient function (§27.2). … ►

§25.15(ii) Zeros

►Since $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L(s,\chi )$ for (the so-called trivial zeros) are as follows: ►

25.15.7 $$L(-2n,\chi )=0\text{ if }\chi (-1)=1,$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $n$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1985b, p. 29)

Permalink:

http://dlmf.nist.gov/25.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

… ►There are also infinitely many zeros in the critical strip $0\le \mathrm{\Re }s\le 1$, located symmetrically about the critical line $\mathrm{\Re }s=\frac{1}{2}$, but not necessarily symmetrically about the real axis. …

32: 32.11 Asymptotic Approximations for Real Variables

… ►Alternatively, if $\nu$ is not zero or a positive integer, then … ►Lastly if $h>h^{\ast }$, then $w_h(x)$ has a simple pole on the real axis, whose location is dependent on $h$. …

33: Bibliography J

… ►

Simple-periodic and Non-periodic Lamé Functions. Mathematical Centre Tracts, No. 72, Mathematisch Centrum, Amsterdam.

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External Links:

ISBN 90-6196-130-0, MathReview (V. P. Madan)

Referenced by:

§29.1, §29.1, §29.18(i), §29.19(i), §29.20(i), §29.20(i), §29.22(ii), §29.3(v), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations L, Notations L, Notations L, Notations L.

Encodings:

BibTeX

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Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:

ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

… ►

On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

…

34: Bibliography S

… ►

A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.

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External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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External Links:

ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

… ►

Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.

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External Links:

ISSN 0012-7094, Link, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

… ►

A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

… ►

The real zeros of Struve’s function. SIAM J. Math. Anal. 1 (3), pp. 365–375.

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External Links:

Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§11.4(vii).

Encodings:

BibTeX

…

35: 3.6 Linear Difference Equations

… ►Difference equations are simple and attractive for computation. … ►The latter method is usually superior when the true value of $w_0$ is zero or pathologically small. …

36: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For $T$ to be actually self adjoint it is necessary to also show that $𝒟(T^{\ast })=𝒟(T)$, as it is often the case that $T$ and $T^{\ast }$ have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator $\frac{d}{dx}$. ►This question may be rephrased by asking: do $f(x)$ and $g(x)$ satisfy the same boundary conditions which are needed to fully specify the solutions of a second order linear differential equation? A simple example is the choice $f(a)=f(b)=0$, and $g(a)=g(b)=0$, this being only one of many. … ►

Example 1: Three Simple Cases where $q(x)=0$, $X=[0,\pi ]$

… ►

§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

… ►this being a matrix element of the resolvent $F(T)={\left(z-T\right)}^{-1}$, this being a key quantity in many parts of physics and applied math, quantum scattering theory being a simple example, see Newton (2002, Ch. 7). …

37: 3.7 Ordinary Differential Equations

… ►For applications to special functions $f$, $g$, and $h$ are often simple rational functions. … ►($𝐈$ and $\mathrm{𝟎}$ being the identity and zero matrices of order $2\times 2$.) … ►The eigenvalues ${\lambda }_k$ are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …

38: 25.16 Mathematical Applications

… ►where the sum is taken over the nontrivial zeros $\rho$ of $\zeta \left(s\right)$. … ► $H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. …

39: 18.39 Applications in the Physical Sciences

… ►Namely the $k$th eigenfunction, listed in order of increasing eigenvalues, starting at $k=0$, has precisely $k$ nodes, as real zeros of wave-functions away from boundaries are often referred to. …Thus the two missing quantum numbers corresponding to EOP’s of order $1$ and $2$ of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range. … ►Here tridiagonal representations of simple Schrödinger operators play a similar role. … ►For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).