doubly-periodic%20forms
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1: 1.13 Differential Equations
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§1.13(vii) Closed-Form Solutions
… ►§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 . Assuming that satisfies un-mixed boundary conditions of the form … ►Transformation to Liouville normal Form
…2: 31.2 Differential Equations
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§31.2(ii) Normal Form of Heun’s Equation
… ►§31.2(iii) Trigonometric Form
… ►§31.2(iv) Doubly-Periodic Forms
►Jacobi’s Elliptic Form
… ►Weierstrass’s Form
…3: 29.12 Definitions
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§29.12(i) Elliptic-Function Form
… ►In consequence they are doubly-periodic meromorphic functions of . … ►The prefixes , , , , , , , 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 every Lamé polynomial in Table 29.12.1 can be written in the form …4: 21.4 Graphics
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5: 22.2 Definitions
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►As a function of , with fixed , each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real.
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6: 20 Theta Functions
Chapter 20 Theta Functions
…7: 26.3 Lattice Paths: Binomial Coefficients
8: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.