Rodrigues%20formula
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11: 36.4 Bifurcation Sets
§36.4(i) Formulas
… ► , cusp bifurcation set: … ► , swallowtail bifurcation set: … ►Elliptic umbilic bifurcation set (codimension three): for fixed , the section of the bifurcation set is a three-cusped astroid … ►Hyperbolic umbilic bifurcation set (codimension three): …12: 18.18 Sums
§18.18(iv) Connection and Inversion Formulas
… ►§18.18(v) Linearization Formulas
… ►Formula (18.18.27) is known as the Hille–Hardy formula. … ►Formula (18.18.28) is known as the Mehler formula. …13: 3.4 Differentiation
Two-Point Formula
… ►Three-Point Formula
… ►Four-Point Formula
… ►Five-Point Formula
… ►Six-Point Formula
…14: 36.5 Stokes Sets
15: 18.39 Applications in the Physical Sciences
16: Bibliography N
17: 25.12 Polylogarithms
18: Bibliography K
19: 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.