Pad%C3%A9%20approximations
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21: Bibliography N
22: 18.40 Methods of Computation
Stieltjes Inversion via (approximate) Analytic Continuation
… βΊResults of low ( to decimal digits) precision for are easily obtained for to . … βΊEquation (18.40.7) provides step-histogram approximations to , as shown in Figure 18.40.1 for and , shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. … βΊIn Figure 18.40.2 the approximations were carried out with a precision of 50 decimal digits.23: 3.4 Differentiation
Laplacian
… βΊBiharmonic Operator
…24: 2.11 Remainder Terms; Stokes Phenomenon
§2.11(i) Numerical Use of Asymptotic Expansions
… βΊ … βΊ§2.11(iii) Exponentially-Improved Expansions
… βΊ§2.11(vi) Direct Numerical Transformations
… βΊFor example, using double precision is found to agree with (2.11.31) to 13D. …25: Bibliography R
26: 28.35 Tables
Blanch and Clemm (1965) includes values of , for , ; , . Also , for , ; , . In all cases . Precision is generally 7D. Approximate formulas and graphs are also included.
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.