As in the mie solution for the sphere, the fields of the incoming plane wave were expanded in terms of the vector wave functions. On the asymptotic expansion of the spheroidal wave function and. Now at university of bucharest, departmentof physics, bucharestmagurele, romania. A computational method to calculate the exact solution for. Spheroidal wave functions become elementary if gammanpi2. These functions 8 mnc, 1, rmnc, result from the solution of the scalar wave equation by separation. Pswf prolate spheroidal wave function acronymfinder. Mar 11, 2016 new relations are found between the spheroidal and spherical wave functions, as well as between the spheroidal functions related to different spheroidal coordinate systems.
Spheroidal wave functions flammer pdf spheroidal wave functions of flammer c. This acclaimed book by carson flammer is available at in several formats for your ereader. Asymptotic expansions of angular prolate spheroidal wave functions for large values of. The spheroidal functions have an important property which makes them very useful in practical work. The package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter gamma. The magic of the prolate spheroidal functions in various setups f. Flammer, spheroidal wave functions, 1957 are included as a special case of meixners more general functions. Figure 1 omitted figure 2 omitted figure 3 omitted figure 4 omitted figure 5 omitted 5 conclusions we have studied properties of prolate spheroidal wave functions pswfs that are bandlimited to 12, 12 and approximately timelimited to t, t for a fixed t 0. This option allows users to search by publication, volume and page selecting this option will search the current publication in context. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. Selecting this option will search all publications across the scitation platform selecting this option will search all publications for the publishersociety in context. Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. Prolate spheroidal wave functions and their properties computation of the pswfs by flammers method uniform estimates of the pswfs and their derivatives applications of the pswfs cimpa school on real and complex analysis with applications, buea cameroun, 114 may 2011. Fourier expansion of generalized prolate spheroidal wave.
Angle spheroidal wave functions prolate spheroidal angle functions sh, 7 are defined to be functions which satisfy the. Cpeanu prolate radial spheroidal wave functions corresponding to this situation is stored in nti and nt2, respectively, and can be printed in the main program. On the asymptotic expansion of the spheroidal wave. The prolate spheroidal wave functions, which are a special case of the spher. Flammer, spheroidal wave functions, 1957 are included as a special case of meixner s more general functions. The prolate angular function of the second kind is given by 8 where is an associated legendre function of the second kind and the coefficients satisfy the recurrence relation. On orthogonal monogenics in oblate spheroidal domains. Prolate spheroidal wave functions of order zero springerlink. Topics include separation of the scalar wave equation in spheroidal coordinates, angle and radial functions, integral representations and relations, and expansions in spherical bessel function products. The property of rayleigh scattering for an anisotropic dielectric ellipsoid authors. The boundary conditions on the surface of the disk determined the scattering components. Spheroidal wave functions dover books on mathematics kindle edition by flammer, carson.
Spheroidal wave functions isbn 9780486446394 pdf epub. Prolate spheroidal wave functions, an introduction to the. The dimension of the space of essentially time and bandlimited signals. For the prolate spheroidal wave function with m0, there exists the integral equation whose kernel is sinxx, and the sinc function kernel sinxx is of great mathematical significance. In mathematics, the prolate spheroidal wave functions pswf are a set of orthogonal bandlimited functions. Universitat regensburg mathematik a spectral representation for. Under these conditions one may ask whether there is a schrodinger equation whose exact eigensolutions are the prolate spheroidal wave functions. Theory and computation of the spheroidal wave functions. Spheroidal wave functions by carson flammer overdrive. As such, they play an important role in signal processing, fourier analysis, and approximation theory. Accurate calculation of oblate spheroidal wave functions arxiv. The wave function of a particle, at a particular time, contains all the information that anybody at that time can have about the particle.
In the paper, we also extend the similar sinc function kernel sinxx to the case m. Pdf theory and computation of spheroidal wavefunctions. Approximation of values of prolate spheroidal wave function. Sections of flammer s work appear in the following brief explanation of the procedure we used to formulate the computer program. Oblatetype asymptotic expansion the asymptotic expansion typically applied to the oblate angular spheroidal wave function oaswf case 3 of eq. The systems must have a common origin of coordinates and a common symmetry axis of coordinate surfaces. Prolate spheroidal wave functions, an introduction to the slepian series and its properties. Spheroidal wave functions flammer pdf writer download. The functions come from the separation of the helmholtz equation in spheroidal coordinates. Prolate radial spheroidal wave functions sciencedirect. Prolate spheroidal wave functions and discrete prolate spheroidal sequences the continuous time problem of maximally concentrating a timelimited function to a limited bandwidth was considered by slepian, pollak, and landau and they found that the prolate spheroidal wave functions pswf are maximally concentrated in this regard 811. Ciric department of electrical and computer engineering university of manitoba winnipeg, manitoba canada r3t sv 6 i.
Recently, pswfs have been becoming increasingly popular in various areas in which such functions occur this includes physics e. Asymptotics of prolate spheroidal wave functions t. Acoustic radiation force exerted on a small spheroidal. The spheroidal wavefunctions of flammer 1957 spheroidal wave functions are included as a special case of meixners more general functions. Bandlimited signal extrapolation using prolate spheroidal. Kaiser windows are in fact generated by sampling this i0sinh approximation. Computation of scattering of a plane wave from multiple. Recent investigations of electromagnetic problems in regions bounded by spheroids have led to an mcreased interest in the spheroidal wave functions. Recurrence relations for prolate spheroidal wave functions.
Abstract not available bibtex entry for this abstract preferred format for this abstract see preferences find. Download it once and read it on your kindle device, pc, phones or tablets. Dunster san diego state university san diego, ca 92182 usa abstract. One of the principal reasons for the importance of pswfs is that they are a. To establish 64 and 67, we combine 29, 32, 33 in section 2. Integral equations for the spinweighted spheroidal wave functions is given. Use features like bookmarks, note taking and highlighting while reading spheroidal wave functions dover books on mathematics. On the evaluation of prolate spheroidal wave functions and. The tables were computed to fiveplace accuracy on the whirlwind digital computer, and automatically tabulated. Prolate spheroidal wave functions pswfs are the eigenfunctions of the bandlimited operator in one dimension. The magic of the prolate spheroidal functions in various.
The magic of the prolate spheroidal functions in various setups. On the asymptotic expansion of the spheroidal wave function. Spheroidal wave functions, stanford university press, stanford, ca, 1957. Uniform asymptotic approximations are obtained for the prolate spheroidal functions ps n mx. Separation of variables for electromagnetic sea ttering by spheroidal particles loan r. Calculation of spheroidal wave functions sciencedirect. Intended to facilitate the use and calculation of spheroidal wave functions, this applicationsoriented text features a. Computation of scattering of a plane wave from multiple prolate spheroids using the collocation multipole method w.
Combining all the results from the previous sections, we can track. Prolate spheroidal wave functions and their properties computation of the pswfs by flammer s method uniform estimates of the pswfs and their derivatives applications of the pswfs wkb method for the pswfs uniform bounds of the pswfs and their derivatives exponential decay of the eigenvalues associated with the pswfs. Spheroidal wave functions dover books on mathematics. Flammer summarizes the work up to 1957 19 and documents the. It appears then logic to investigate the existence of a schrodinger equation which admits the pswfs as exact analytical solutions. An introduction discusses the mathematical properties of the functions and describes some of their applications.
A set of tables of spheroidal wave functions designed to simplify the computation of acoustic and electromagnetic scattering from spheroids. The applicability ranges of the relations obtained are discussed. Integral equations for the spinweighted spheroidal wave. A significant step in the process is the higher order piecewise polynomial approximation of the overlap integral required for obtaining the expansion coefficients accurately with very high precision. Flammer, spheroidal wave functions dover publications, mineola, ny, 2005. Spheroidal wave functions, integral equation, green function 1. Separation of variables of the wave equation in spheroidal coordinates results in a differential equation that is satis. The prolate radial spheroidal wave functions appear in a wide press, stanford, 1957 russian translation, 1962. The text concludes with an appendix, references, and tables of. We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for.
Explicit upper bounds on the eigenvalues associated with. Prolate spheroidal wave functions of order zero download prolate spheroidal wave functions of order zero ebook pdf or read online books in pdf, epub, and mobi format. Rokhlin, highfrequency asymptotic expansions for certain prolate spheroidal wave functions, the journal of fourier analysis and applications, 9 2003. The singular functions for the problem of recovering a time limited function from its fourier transform in a certain band of frequencies are given, in the simplest case, by the prolate spheroidal wave functions. Intended to facilitate the use and calculation of spheroidal wave functions, this applicationsoriented text features a detailed and unified account of the properties of these functions.
We show that the theory of the osqwfs is determined by the moisilteodorescu. Computational geometric and boundary value properties of. Spheroidaleigenvaluenu,mu,gamma 11 formulas primary definition 1 formula specific values 6 formulas. These charts show that the eigenvalues have many branch points in the complex planes of their arguments.
Alternative expressions for calculating the oblate spheroidal radial functions of both kinds 1, ml. Spheroidaleigenvaluenu,mu,gamma 11 formulas primary definition 1 formula specific values 6 formulas general characteristics 4 formulas spheroidaleigenvalue nu, mu, gamma 11 formulas spheroidaleigenvalue. Eigenvalues of spheroidal wave functions and their branch. Buy spheroidal wave functions dover books on mathematics on free shipping on qualified orders. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and bandlimit. Eigenvalues of spheroidal wave functions are calculated as functions of complex propagation constants.
Rotationaltranslational addition theorems for the scalar spheroidal wave. The functions ri and r2 end when the relative accuracy errb is achieved. Read spheroidal wave functions by carson flammer available from rakuten kobo. The spheroidal wavefunctions of flammer 1957 spheroidal wave functions are included as a special case of meixners. Theory and computation of the spheroidal wave functions core. Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it presents tables that provide a. Jul 06, 2015 what is a prolate spheroid wave function and how does it apply to eegs and brain mapping. Additional subjects include recurrence relations of whittaker type, asymptotic expansions for large values of c, and vector wave functions. Cartesian derivatives of the scalar spheroidal wave functions are derived, leading. Download pdf prolate spheroidal wave functions of order. These are of great utility in disciplines such as geodesy or cosmology. Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it presents tables that provide a convenient means for handling wave problems in spheroidal coordinates.
For the probability interpretation to make sense, the. We introduce analogous techniques based on the assumption that the function to be dealt with is bandlimited, and use the well developed apparatus of prolate spheroidal wavefunctions to construct quadratures, interpolation and differentiation formulae, etc, for bandlimited functions. Obtained by flammer spheroidal wave functions stanford university press. There are many fields where spheroidal functions play important roles just as the spherical functions do. Spheroidal wave functions are general izations of legendre functions. The wave equation in spheroidal coordinates and its. Prolate spheroidal wave functions whose domain is a portion of the surface of the unit sphere are more generally called slepian functions see also spectral concentration problem. This expansion formula being of very general nature can be transformed to provide many new results involving various commonly used special functions occuring in applied mathematics, mathematics physics and mchanics. A new integral equation for the spheroidal equations in.
Prolate spheroidal wave function as exact solution of the. Some relations between the spheroidal and spherical wave. Intended to facilitate the use and calculation of spheroidal wave functions. Yingle li, jin li, mingjun wang, qunfeng dong keywords. On the use of discrete prolate spheroidal windows for. Prolate spheroidal wave functions and applications. Prolate spheroidal wave function how is prolate spheroidal. Miranian department of mathematics, university of california, berkeley ca, 94720 abstract the singular functions for the problem of recovering a time limited function from its fourier transform in a certain. Prolate spheroidal wave functions, fourier analysis and uncertainty iv. Click download or read online button to prolate spheroidal wave functions of order zero book pdf. There are two kinds of eigenvalues in the differential and corresponding integral equations, and the relation between them is given explicitly. Spheroidal wave functions are solutions of the helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical harmonics.
They are of interest because spheroids give tractable models for various boundaryvalue problems in electromagnetism and acoustics. Prolate spheroidal wave functions pswfs play an important role in. Separation of variables for electromagnetic sea ttering by. Enter your mobile number or email address below and well send you a link to download the free kindle app.
Spheroidal wave functions dover books on mathematics and millions of other books are available for amazon kindle. During the course of finding, we establish several particular cases. Theory and computation of the spheroidal wave functions arxiv. This paper introduces the oblate spheroidal quaternionic wave functions osqwfs, which extend the oblate spheroidal wave functions introduced in the late 1950s by c. Flammer, spheroidal wave functions, stanford university press. Chen2 1department of mechanical engineering, china university of science and technology, taipei, taiwan 2department of harbor and river engineering, national taiwan ocean university, keelung, taiwan. Shortly after meixner, flammer 7 derived another solution using oblate spheroidal vector wave functions to expand the fields. Prolate spheroidal wave functions, fourier analysis and uncertainty iii. Theory and computation of spheroidal wavefunctions. The spheroidal wavefunctions of flammer 1957 spheroidal wave functions. This paper discusses the calculation of spheroidal wave functions by direct numerical methods. Introduction the spheroidal wave equations are extension of the ordinary spherical equations.
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