relation to Lamé equation

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1: 29.2 Differential Equations

… ►For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). …

2: 31.8 Solutions via Quadratures

… ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …

3: 29.12 Definitions

§29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►There are eight types of Lamé polynomials, defined as follows: …In consequence they are doubly-periodic meromorphic functions of $z$. … ►

§29.12(ii) Algebraic Form

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4: Bibliography

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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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External Links:

MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§23.21(ii).

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:

ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Tables of Lamé Polynomials. Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§29.15(ii), 2nd item.

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.

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5: 31.7 Relations to Other Functions

§31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

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§31.7(ii) Relations to Lamé Functions

… ►equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

6: 29.6 Fourier Series

§29.6 Fourier Series

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§29.6(i) Function ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$

… ►In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. … ►Consequently, ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$ reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). … ►

§29.6(ii) Function ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$

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7: Bibliography S

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

Encodings:

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Lamé polynomial solutions to some elliptic crack and punch problems. Internat. J. Engrg. Sci. 16 (8), pp. 551–563.

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External Links:

ISSN 0020-7225, Document, MathReview, ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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External Links:

ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

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The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).

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Notes:

In Russian. English translation: Differential Equations 23(1987), no. 5, pp. 569–576.

External Links:

ISSN 0374-0641, MathReview (T. Chanturiya), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

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8: 28.34 Methods of Computation

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§28.34(i) Characteristic Exponents

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Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

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Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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9: Errata

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Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

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Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_k=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}{u}_{k-1}$ was added to Equation (9.7.2).

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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10: 29.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. ►The main functions treated in this chapter are the eigenvalues $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$, and the Lamé polynomials ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$. … ►Other notations that have been used are as follows: Ince (1940a) interchanges $a_{\nu }^{2m+1}\left(k^2\right)$ with $b_{\nu }^{2m+1}\left(k^2\right)$. The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by …The relation to the Lamé functions ${\mathrm{Ec}}_{\nu }^m$, ${\mathrm{Es}}_{\nu }^m$ of Ince (1940b) is given by …