relations to hyperbolic functions
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1: 4.28 Definitions and Periodicity
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Relations to Trigonometric Functions
…2: 10.39 Relations to Other Functions
§10.39 Relations to Other Functions
►Elementary Functions
… ►Parabolic Cylinder Functions
… ►Confluent Hypergeometric Functions
… ►Generalized Hypergeometric Functions and Hypergeometric Function
…3: 19.10 Relations to Other Functions
§19.10 Relations to Other Functions
►§19.10(i) Theta and Elliptic Functions
►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►§19.10(ii) Elementary Functions
… ►For relations to the Gudermannian function and its inverse (§4.23(viii)), see (19.6.8) and …4: 13.18 Relations to Other Functions
§13.18 Relations to Other Functions
►§13.18(i) Elementary Functions
… ►§13.18(iv) Parabolic Cylinder Functions
… ►§13.18(v) Orthogonal Polynomials
… ►Laguerre Polynomials
…5: 22.16 Related Functions
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22.16.8
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6: 28.32 Mathematical Applications
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►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient.
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►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting , in (28.32.3)).
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►approaches the same value when tends to the endpoints of .
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►Two conditions are used to determine .
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7: 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. … ►A related hyperbolic series is ►
22.11.14
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8: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
►§14.19(i) Introduction
… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates , which are related to Cartesian coordinates by … ►§14.19(iv) Sums
… ►§14.19(v) Whipple’s Formula for Toroidal Functions
…9: 36.2 Catastrophes and Canonical Integrals
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