Jacobian
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11: 22.5 Special Values
§22.5 Special Values
… βΊTable 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its -derivative (or at a pole, the residue), for values of that are integer multiples of , . … βΊ βΊ§22.5(ii) Limiting Values of
… βΊ12: 22.13 Derivatives and Differential Equations
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Table 22.13.1: Derivatives of Jacobian elliptic functions with respect to variable.
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§22.13(i) Derivatives
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§22.13(ii) First-Order Differential Equations
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22.13.7
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§22.13(iii) Second-Order Differential Equations
…13: 22.10 Maclaurin Series
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§22.10(i) Maclaurin Series in
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22.10.1
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§22.10(ii) Maclaurin Series in and
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22.10.6
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14: 22 Jacobian Elliptic Functions
Chapter 22 Jacobian Elliptic Functions
…15: 22.3 Graphics
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§22.3(i) Real Variables: Line Graphs
… βΊ§22.3(iii) Complex ; Real
… βΊ§22.3(iv) Complex
… βΊ βΊ …16: 22.19 Physical Applications
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§22.19(ii) Classical Dynamics: The Quartic Oscillator
… βΊ§22.19(iii) Nonlinear ODEs and PDEs
… βΊ§22.19(iv) Tops
… βΊ§22.19(v) Other Applications
… βΊ17: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.2
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22.12.8
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22.12.11
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22.12.13
18: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate
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βΊJacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena.
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19: 22.11 Fourier and Hyperbolic Series
§22.11 Fourier and Hyperbolic Series
… βΊIn (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. … βΊ
22.11.13
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βΊA related hyperbolic series is
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