Elliptic and Spheroidal Wave Functions

Chu, Lan Jen; Stratton, J. A.

doi:10.1002/sapm1941201259

Cited by 30 publications

(15 citation statements)

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“…1. Many researchers have obtained the electromagnetic field vectors inside and outside the oblate spheroids with various methods, such as analytical and infinite element methods [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In the present study, we obtained the calculation results in the oblate coordinate system (g, n, /) using the analytical method used by Asano [5,12].…”

Section: Scattering Field Vectorsmentioning

confidence: 98%

Optical characteristics of flowing blood

et al. 2011

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The transmitted light strength (TS) through a thin blood layer changes with variation in blood flow, such as positive streaming transparency for low hematocrits and negative streaming transparency for high hematocrits. These phenomena are examined theoretically and experimentally. Maxwell's equations are solved assuming that erythrocytes are oblate spheroids to investigate these phenomena due to flowing blood. The theoretical results reveal that the scattering and absorption cross sections for flowing blood are larger than those for stagnant blood. Experimental results indicate that the TS for both oxygenated and deoxygenated flowing blood, with a hematocrit of up to approximately 20%, was stronger than that for stagnant blood. The TS decreased for flowing blood with a hematocrit of approximately 20% or greater. Applying the theoretical scattering and absorption cross sections to the absorption and multiple scattering theory of Victor Twersky, the changes in the TS due to flowing blood are obtained theoretically. From the theoretical and experimental results, the positive streaming transparency phenomenon of flowing blood with a low hematocrit and the negative streaming transparency phenomenon with a high hematocrit are found to result from increased scattering and absorption cross sections because of the orientation of flowing erythrocytes.

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Section: Scattering Field Vectorsmentioning

confidence: 98%

Optical characteristics of flowing blood

et al. 2011

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“…-1 r=O h 2r dt 2r By direct evaluation of the bracketed expression one can show, with the help of (6), that it is proportional to Smn(h, t), at least for the first few orders. From thIS the orthogonality property (23) follows directly.…”

Section: )Mi2s Mn (H T)]} Smp(h T) Dtmentioning

confidence: 98%

“…[9, [4] for the MathIeU functIOns, but It appears to remain undeveloped for the spherOIdal wave functions. The prInCIpal sources on the propertIes summarIzed above for the general spheroidal functIOns are Poole [11], Stratton [14], and Chu and Stratton [6]. In addItIOn to these, the prinCIpal sources on the spherOIdal wave functions are MeIxner and Schafke [8] and Morse and Feshbach [9], and for the perIodic Mathieu functIOns Blanch [4], MeIxner and Schafke [8], Morse and Feshbach [9], Poole [10], and Stratton [15, pp.…”

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confidence: 99%

On Some Double Orthogonality Properties of the Spheroidal and Mathieu Functions

Rhodes

1965

Journal of Mathematics and Physics

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1.Introduction. The existence of complete sets of functions that are orthogonal over two different regions simultaneously, one contained within the other, appears to have been discovered by Bergman [1,2,3]. They were applied later by Davis [7] to the problem of finding the best mean-square approximation to a function given over the smaller region when a constraint is imposed upon the mean-square value of the approximating function over the larger region. In the theory of doubly orthogonal functions as developed by Bergman and Davis the regions of orthogonality are areas in the complex plane. More recently Sleplan and Pollak [13] have shown that the zero-order prolate spheroidal wave functions are doubly orthogonal over two dIfferent intervals along the real axis. These functions have been found to be of considerable importance in certain physical problems, namely in the representation of signals [13] and in the synthesis of line source antennas [12]. Further investigation into the antenna synthesis problem has led to the existance of a broader class of doubly orthogonal functions, a class that includes all of the first kind of spheroidal wave functions and periodic Mathieu functions.

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“…In Section 2 we deal with the two-sided infinite expansions for CHE (1), its limiting and particular cases. For the spheroidal equation, we get the Meixner solutions in series of Bessel functions [9] instead of the Chu and Stratton solutions [16] mentioned by Leaver. In Section 3 we apply the analysis of section 2 to one-sided expansions. These are generated by truncating on the left-hand side the two-sided series: by requiring that n ≥ 0 we determine the parameter ν of the two-sided solutions as function of the parameters of the differential equations.…”

Section: Introductionmentioning

confidence: 99%

Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions

El-Jaick¹,

Figueiredo²

2013

J. Phys. A: Math. Theor.

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The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index n runs from minus to plus infinity [E. W. Leaver, J. Math. Phys. 27, 1238Phys. 27, (1986]. First we show that, in contrast to the D'Alembert test, under certain conditions the Raabe test assures that the domains of convergence of these solutions include an additional singular point. Further, by using a limit proposed by Leaver, we obtain solutions for the double-confluent Heun equation (DCHE). In addition, we get solutions for the so-called Whittaker-Ince limit of the CHE and DCHE. For these four equations, new solutions are generated by transformations of variables. In the second place, for each of the above equations we obtain one-sided series solutions (n ≥ 0) by truncating on the left the two-sided series. Finally we discuss the time dependence of the Klein-Gordon equation in two cosmological models and the spatial dependence of the Schrödinger equation to a family of quasi-exactly solvable potentials. For a subfamily of these potentials we obtain infinite-series solutions which converge and are bounded for all values of the independent variable, in opposition to a common belief.

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Elliptic and Spheroidal Wave Functions

Cited by 30 publications

References 0 publications

Optical characteristics of flowing blood

Optical characteristics of flowing blood

On Some Double Orthogonality Properties of the Spheroidal and Mathieu Functions

Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions

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