doubly-periodic forms

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1: 1.13 Differential Equations

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§1.13(vii) Closed-Form Solutions

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►This is the Sturm-Liouville form of a second order differential equation, where ^′ denotes $\frac{d}{dx}$. Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form … ►

Transformation to Liouville normal Form

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2: 31.2 Differential Equations

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§31.2(ii) Normal Form of Heun’s Equation

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§31.2(iii) Trigonometric Form

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§31.2(iv) Doubly-Periodic Forms

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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3: 29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►In consequence they are doubly-periodic meromorphic functions of $z$. … ►The prefixes $u$, $s$, $c$, $d$, $\mathrm{𝑠𝑐}$, $\mathrm{𝑠𝑑}$, $\mathrm{𝑐𝑑}$, $\mathrm{𝑠𝑐𝑑}$ indicate the type of the polynomial form of the Lamé polynomial; compare the 3rd and 4th columns in Table 29.12.1. … ►

§29.12(ii) Algebraic Form

►With the substitution $\xi ={\mathrm{sn}}^2(z,k)$ every Lamé polynomial in Table 29.12.1 can be written in the form …

4: 21.4 Graphics

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Figure 21.4.4: A real-valued scaled Riemann theta function: $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|{𝛀}_1)$, $0\le x\le 4$, $0\le y\le 4$. In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration. Magnify 3D Help

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5: 22.2 Definitions

… ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. …

6: 31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

7: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

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8: 31.12 Confluent Forms of Heun’s Equation

§31.12 Confluent Forms of Heun’s Equation

►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. …There are four standard forms, as follows: … ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►

9: Frank Garvan

… ►His research is in the areas of $q$-series and modular forms, and he enjoys using MAPLE in his research. …

10: Bruce R. Miller

… ►He is the developer of the tools used to process the DLMF into both book and web forms. …