eigenvalue%2Feigenvector%20characterization
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11—20 of 823 matching pages
11: 28.35 Tables
Blanch and Rhodes (1955) includes , , , ; 8D. The range of is 0 to 0.1, with step sizes ranging from 0.002 down to 0.00025. Notation: , .
Ince (1932) includes eigenvalues , , and Fourier coefficients for or , ; 7D. Also , for , , corresponding to the eigenvalues in the tables; 5D. Notation: , .
National Bureau of Standards (1967) includes the eigenvalues , for with , and with ; Fourier coefficients for and for , , respectively, and various values of in the interval ; joining factors , for with (but in a different notation). Also, eigenvalues for large values of . Precision is generally 8D.
Zhang and Jin (1996, pp. 521–532) includes the eigenvalues , for , ; (’s) or 19 (’s), . Fourier coefficients for , , . Mathieu functions , , and their first -derivatives for , . Modified Mathieu functions , , and their first -derivatives for , , . Precision is mostly 9S.
Blanch and Clemm (1969) includes eigenvalues , for , , , ; 4D. Also and for , , and , respectively; 8D. Double points for ; 8D. Graphs are included.
12: 29.7 Asymptotic Expansions
§29.7(i) Eigenvalues
… ►13: 28.2 Definitions and Basic Properties
§28.2(v) Eigenvalues ,
►For given and , equation (28.2.16) determines an infinite discrete set of values of , the eigenvalues or characteristic values, of Mathieu’s equation. … ►Distribution
… ►Change of Sign of
… ►Table 28.2.2 gives the notation for the eigenfunctions corresponding to the eigenvalues in Table 28.2.1. …14: 28.12 Definitions and Basic Properties
§28.12(i) Eigenvalues
… ►For given (or ) and , equation (28.2.16) determines an infinite discrete set of values of , denoted by , . …For other values of , is determined by analytic continuation. … … ►As a function of with fixed (), is discontinuous at . …15: 30.3 Eigenvalues
§30.3 Eigenvalues
… ►These solutions exist only for eigenvalues , , of the parameter . … ►The eigenvalues are analytic functions of the real variable and satisfy … ►has the solutions , . If is an odd positive integer, then Equation (30.3.5) has the solutions , . …16: 30.16 Methods of Computation
§30.16(i) Eigenvalues
… ►and real eigenvalues , , , , arranged in ascending order of magnitude. … ►which yields . … ►If is known, then can be found by summing (30.8.1). … ►Form the eigenvector of associated with the eigenvalue , , normalized according to …17: 30.18 Software
18: 30.9 Asymptotic Approximations and Expansions
§30.9 Asymptotic Approximations and Expansions
… ►As , with , … ►The asymptotic behavior of and as in descending powers of is derived in Meixner (1944). …The behavior of for complex and large is investigated in Hunter and Guerrieri (1982).19: 30.17 Tables
§30.17 Tables
►Stratton et al. (1956) tabulates quantities closely related to and for , , . Precision is 7S.
Flammer (1957) includes 18 tables of eigenvalues, expansion coefficients, spheroidal wave functions, and other related quantities. Precision varies between 4S and 10S.
Van Buren et al. (1975) gives , for , , . Precision is 8S.
Zhang and Jin (1996) includes 24 tables of eigenvalues, spheroidal wave functions and their derivatives. Precision varies between 6S and 8S.