entire functions

SearchAdvancedHelp

(0.004 seconds)

31—40 of 48 matching pages

31: 1.10 Functions of a Complex Variable

… ►is an entire function with zeros at $z_n$. …

32: Bibliography B

… ►

Coulomb functions in entire ($\eta ,\rho$)-plane. Comput. Phys. Comm. 3 (2), pp. 73–87.

ⓘ

External Links:

Document, Code

Referenced by:

§33.23(vi).

Encodings:

BibTeX

…

33: 10.2 Definitions

… ►When $\nu =n$ $(\in \mathbb{Z})$, $J_{\nu }\left(z\right)$ is entire in $z$. ►For fixed $z$ $(\ne 0)$ each branch of $J_{\nu }\left(z\right)$ is entire in $\nu$. … ►For fixed $z$ $(\ne 0)$ each branch of $Y_{\nu }\left(z\right)$ is entire in $\nu$. … ►For fixed $z$ $(\ne 0)$ each branch of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ is entire in $\nu$. … ►

Cylinder Functions

…

34: 4.14 Definitions and Periodicity

… ►The functions $\sin z$ and $\cos z$ are entire. …

35: Errata

… ►

The following additions were made in Chapter 1:

• New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).

• The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).

• In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).

• In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).

• New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).

• New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).

• Another form of Parseval’s formula, (1.14.7_5).

• We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.

• Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.

• An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

… ►

The original constraint, $\mathrm{\Re }s>0$, was removed because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta \left(s\right)-{(s-1)}^{-1}$ is an entire function.

Suggested by John Harper.

…

36: 23.2 Definitions and Periodic Properties

… ►The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. …

37: 10.25 Definitions

… ►Its solutions are called modified Bessel functions or Bessel functions of imaginary argument. ►

§10.25(ii) Standard Solutions

… ►For fixed $z$ $(\ne 0)$ each branch of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ is entire in $\nu$. ►

Branch Conventions

…

38: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. In particular, asymptotic formulas in terms of elementary functions are given when $z=x$ is real and fixed. … ►This expansion is in terms of the parabolic cylinder function and its derivative. … ►This expansion is in terms of confluent hypergeometric functions. … ►Both expansions are in terms of parabolic cylinder functions. …

39: 18.39 Applications in the Physical Sciences

… ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. …

40: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►If $b_1=b_2=0$, then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$-space. …