On the completeness of the spherical vector wave functions

Aydin, K.; Hizal, A.

doi:10.1016/0022-247x(86)90233-7

Cited by 28 publications

(16 citation statements)

References 6 publications

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“…In the dipole's layer

V_{q}

, the secondary component

{\tilde{boldG}}^{\sec}

of the Green's function is expanded according to the right‐hand side of with

p = q

. When

k_{p}

is not an eigenvalue of the homogeneous Dirichlet problem for in

V_{p}

, then the series converges uniformly in the closed subsets of

V_{p}

and in the mean square sense on the boundary of

V_{p}

.…”

Section: Exact Dyadic Green's Functionmentioning

confidence: 99%

“…The spherical vector wave functions constitute an orthogonal set ((10.11)-(10.13) of [30]). The uniform convergence of the series (11) was described in [18] by following the analysis of [35]. The secondary components G p of the Green's function in the spherical shells V p ( p = 1, .…”

Section: Exact Dyadic Green's Functionmentioning

confidence: 99%

“…Similarly, the 0-excitation far-field pattern of a homogeneous dielectric ball due to an arbitrary external dipole is given by (35) while the 1-excitation far-field pattern of a homogeneous dielectric ball due to an arbitrary internal dipole is given by…”

Section: Electric Fields and Far-field Patternsmentioning

confidence: 99%

“…In the framework of inverse scattering algorithms, it is difficult to recover the characteristics of the spherical medium and the dipole by using the exact far-field patterns (35) and (36), derived in Section 4. However, under the low-frequency assumption k 0 a 1 1, namely, by assuming that the ball's radius a 1 is much smaller than the primary field's wavelength, we may obtain simplified expressions of the far-field patterns.…”

Section: Low-frequency Far-field Patternsmentioning

confidence: 99%

See 3 more Smart Citations

Electromagnetic Excitation of a Spherical Medium by an Arbitrary Dipole and Related Inverse Problems

Prokopiou

Tsitsas

2018

Stud Appl Math

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A piecewise homogeneous spherical medium is excited by an external or internal electric dipole with arbitrary location and polarization. The dyadic Green's function of the medium is determined analytically. Then, the vector electric fields and far‐field patterns are obtained. Low‐frequency approximations of the far‐field patterns are subsequently derived, which encode the dipole's locations coordinates and polarization components in the different orders of the associated expansions. This fact enables the establishment of far‐field inverse scattering algorithms referring to the electromagnetic interior or exterior excitation of a small sphere by an arbitrary dipole. Inverse medium and inverse source problems are considered concerning, respectively, the determination of the scatterer's material parameters and the dipole's characteristics. The developed inverse algorithms determine exactly the unknown parameters of problems fulfilling the low‐frequency assumption, which is indeed the case in most relevant applications, like, e.g., in biomedical imaging.

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“…In the dipole's layer

V_{q}

, the secondary component

{\tilde{boldG}}^{\sec}

of the Green's function is expanded according to the right‐hand side of with

p = q

. When

k_{p}

is not an eigenvalue of the homogeneous Dirichlet problem for in

V_{p}

, then the series converges uniformly in the closed subsets of

V_{p}

and in the mean square sense on the boundary of

V_{p}

.…”

Section: Exact Dyadic Green's Functionmentioning

confidence: 99%

Section: Exact Dyadic Green's Functionmentioning

confidence: 99%

Section: Electric Fields and Far-field Patternsmentioning

confidence: 99%

Section: Low-frequency Far-field Patternsmentioning

confidence: 99%

See 2 more Smart Citations

Electromagnetic Excitation of a Spherical Medium by an Arbitrary Dipole and Related Inverse Problems

Prokopiou

Tsitsas

2018

Stud Appl Math

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“…The general theory and background of the SWFs are contained in the monograph by W. W. Bell [3] (cf. [4,5]).…”

Section: The Spherical Wave Functionsmentioning

confidence: 99%

Quaternionic spherical wave functions

Morais

Le²,

Rosa

2016

Math Methods in App Sciences

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The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu‐type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti‐hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of D0‐hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.

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Bibliography

2010

Foundations of Applied Electrodynamics

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On the completeness of the spherical vector wave functions

Cited by 28 publications

References 6 publications

Electromagnetic Excitation of a Spherical Medium by an Arbitrary Dipole and Related Inverse Problems

Electromagnetic Excitation of a Spherical Medium by an Arbitrary Dipole and Related Inverse Problems

Quaternionic spherical wave functions

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