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1: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids

►Equation (30.13.7) for $\xi \le {\xi }_0$ together with the boundary condition $w=0$ on the ellipsoid given by $\xi ={\xi }_0$, poses an eigenvalue problem with ${\kappa }^2$ as spectral parameter. …

2: 30.13 Wave Equation in Prolate Spheroidal Coordinates

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§30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids

►Equation (30.13.7) for $\xi \le {\xi }_0$, and subject to the boundary condition $w=0$ on the ellipsoid given by $\xi ={\xi }_0$, poses an eigenvalue problem with ${\kappa }^2$ as spectral parameter. …For the Dirichlet boundary-value problem of the region ${\xi }_1\le \xi \le {\xi }_2$ between two ellipsoids, the eigenvalues are determined from …

3: 14.31 Other Applications

… ►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). … ►

§14.31(iii) Miscellaneous

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4: Bibliography

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High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder. J. Opt. Soc. Amer. A 14 (6), pp. 1305–1315.

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§36.14(ii).

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Unsteady lifting-line theory as a singular-perturbation problem. J. Fluid Mech 153, pp. 59–81.

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External Links:

Document, ZentralBlatt

Referenced by:

§11.12.

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Note on the trivial zeros of Dirichlet $L$-functions. Proc. Amer. Math. Soc. 94 (1), pp. 29–30.

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ISSN 0002-9939, FullText, MathReview (Jerzy Kaczorowski), ZentralBlatt

Referenced by:

(25.15.7), (25.15.8), §25.15(ii), §25.15.

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Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.

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ISBN 0-387-97127-0, MathReview, ZentralBlatt

Referenced by:

§17.2(i), §23.10(iv), §23.15(i), §23.17(ii), §23.18, §23.18, §23.19, §23.20(i), §23.20(v), Ch.23, §27.14(iii), §27.14(iv), §27.14(vi), §27.7, Ch.27.

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5: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►In other words, Dirichlet characters (mod $k$) satisfy the four conditions: … ►If $\chi$ is a character (mod $k$), so is its complex conjugate $\overline{\chi }$. … ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if … ►Every Dirichlet character $\chi$ (mod $k$) is a product …

6: 23.20 Mathematical Applications

… ►The interior of $R$ is mapped one-to-one onto the lower half-plane. … ►The interior of the rectangle with vertices $0$, ${\omega }_1$, $2{\omega }_3$, $2{\omega }_3-{\omega }_1$ is mapped two-to-one onto the lower half-plane. The interior of the rectangle with vertices $0$, ${\omega }_1$, $\frac{1}{2}{\omega }_1+{\omega }_3$, $\frac{1}{2}{\omega }_1-{\omega }_3$ is mapped one-to-one onto the lower half-plane with a cut from $e_3$ to $\mathrm{\wp }\left(\frac{1}{2}{\omega }_1+{\omega }_3\right)(=\mathrm{\wp }\left(\frac{1}{2}{\omega }_1-{\omega }_3\right))$. … ►For applications of the Weierstrass function and the elliptic curve method to these problems see Bressoud (1989) and Koblitz (1999). …

7: 25.15 Dirichlet $L$-functions

§25.15 Dirichlet $L$-functions

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§25.15(i) Definitions and Basic Properties

►The notation $L(s,\chi )$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … … ►

§25.15(ii) Zeros

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8: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). ►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta ,\mathrm{\ell }+\frac{1}{2}}\left(2\rho \right)$. … ►

9: 2.4 Contour Integrals

… ►We assume that in any closed sector with vertex $t=0$ and properly interior to , the expansion (2.3.7) holds as $t\to 0$, and $q(t)=O\left({\mathrm{e}}^{\sigma |t|}\right)$ as $t\to \mathrm{\infty }$, where $\sigma$ is a constant. Then (2.4.1) is valid in any closed sector with vertex $z=0$ and properly interior to . … ►Now suppose that in (2.4.10) the minimum of $\mathrm{\Re }\left(zp(t)\right)$ on $𝒫$ occurs at an interior point $t_0$. … ►In the commonest case the interior minimum $t_0$ of $\mathrm{\Re }\left(zp(t)\right)$ is a simple zero of $p^{\prime }(t)$. … ►The problem of obtaining an asymptotic approximation to $I(\alpha ,z)$ that is uniform with respect to $\alpha$ in a region containing $\widehat{\alpha }$ is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). …

10: 27.11 Asymptotic Formulas: Partial Sums

§27.11 Asymptotic Formulas: Partial Sums

… ►For example, Dirichlet (1849) proves that for all $x\ge 1$, …Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number ${\theta }_0$ such that the error term in (27.11.2) is $O\left(x^{\theta }\right)$ for all $\theta >{\theta }_0$. … ►where $\left(h,k\right)=1$, $k>0$. ►Letting $x\to \mathrm{\infty }$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h(modk)$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. …