Helmholtz%20equation
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11: 12.17 Physical Applications
§12.17 Physical Applications
►The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation …The first two equations can be transformed into (12.2.2) or (12.2.3). ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. …12: 28.32 Mathematical Applications
§28.32(ii) Paraboloidal Coordinates
… ►When the Helmholtz equation …13: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…14: 20 Theta Functions
Chapter 20 Theta Functions
…15: 28.16 Asymptotic Expansions for Large
16: Bibliography O
17: Bibliography N
18: 3.8 Nonlinear Equations
19: 28.18 Integrals and Integral Equations
§28.18 Integrals and Integral Equations
…20: 28.35 Tables
§28.35 Tables
… ►Ince (1932) includes eigenvalues , , and Fourier coefficients for or , ; 7D. Also , for , , corresponding to the eigenvalues in the tables; 5D. Notation: , .
Kirkpatrick (1960) contains tables of the modified functions , for , , ; 4D or 5D.
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.