finite%20Fourier%20series
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1: 29.20 Methods of Computation
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βΊSubsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6.
…The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii).
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βΊA third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv).
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βΊThe eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998).
The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials.
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2: Bibliography R
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Fourier analysis and signal processing by use of the Möbius inversion formula.
IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.
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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness.
Academic Press, New York.
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On the foundations of combinatorial theory. VIII. Finite operator calculus.
J. Math. Anal. Appl. 42, pp. 684–760.
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Finite-sum rules for Macdonald’s functions and Hankel’s symbols.
Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
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3: Peter L. Walker
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βΊWalker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004.
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4: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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Finite Differences and Difference Equations in the Real Domain.
Clarendon Press, Oxford.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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5: 20 Theta Functions
Chapter 20 Theta Functions
…6: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation.
Adv. Quantum Chem. 72, pp. 95–127.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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7: 36.15 Methods of Computation
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§36.15(i) Convergent Series
βΊClose to the origin of parameter space, the series in §36.8 can be used. … βΊ§36.15(iv) Integration along Finite Contour
βΊThis can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …8: Bibliography W
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Tables of Summable Series and Integrals Involving Bessel Functions.
Holden-Day, San Francisco, CA.
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Some transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
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On the asymptotic behavior of the Fourier coefficients of Mathieu functions.
J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
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Asymptotic expansions of Fourier transforms of functions with logarithmic singularities.
J. Math. Anal. Appl. 64 (1), pp. 173–180.
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9: 27.10 Periodic Number-Theoretic Functions
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βΊEvery function periodic (mod ) can be expressed as a finite Fourier
series of the form
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βΊis a periodic function of and has the finite Fourier-series expansion
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βΊThe finite Fourier expansion of a primitive Dirichlet character has the form
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10: Bibliography C
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Introduction to the Theory of Fourier’s Series and Integrals.
3rd edition, Macmillan, London.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Advanced Combinatorics: The Art of Finite and Infinite Expansions.
enlarged edition, D. Reidel Publishing Co., Dordrecht.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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