relations to Jacobian elliptic functions
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11: 22.16 Related Functions
12: 22.11 Fourier and Hyperbolic Series
§22.11 Fourier and Hyperbolic Series
… ►In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. … ►Similar expansions for and follow immediately from (22.6.1). … ►A related hyperbolic series is …13: 20.11 Generalizations and Analogs
14: 29.6 Fourier Series
§29.6 Fourier Series
… ►With , as in (29.2.5), we have … ►In addition, if satisfies (29.6.2), then (29.6.3) applies. … ►Consequently, reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). … ►Here is as in §22.2, and …15: 22.20 Methods of Computation
§22.20 Methods of Computation
… ►§22.20(iii) Landen Transformations
… ►§22.20(iv) Lattice Calculations
… ►§22.20(v) Inverse Functions
… ►16: William P. Reinhardt
17: 23.21 Physical Applications
§23.21 Physical Applications
… ►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form . … ►§23.21(ii) Nonlinear Evolution Equations
►Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … ►§23.21(iii) Ellipsoidal Coordinates
…18: 22.8 Addition Theorems
§22.8 Addition Theorems
… ►§22.8(iii) Special Relations Between Arguments
… ►For these and related identities see Copson (1935, pp. 415–416). ►If sums/differences of the ’s are rational multiples of , then further relations follow. …19: Bibliography W
20: Software Index
These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.
Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.
A cross index of mathematical software in use at NIST.