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1: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity
2: 31.2 Differential Equations
§31.2(i) Heun’s Equation
Jacobi’s Elliptic Form
Weierstrasss Form
where 2 ω 1 and 2 ω 3 with ( ω 3 / ω 1 ) > 0 are generators of the lattice 𝕃 for ( z | 𝕃 ) . …
§31.2(v) Heun’s Equation Automorphisms
3: 29.2 Differential Equations
§29.2(i) Lamé’s Equation
§29.2(ii) Other Forms
we have …For the Weierstrass function see §23.2(ii). …
4: 7.20 Mathematical Applications
§7.20(ii) Cornu’s Spiral
Let the set { x ( t ) , y ( t ) , t } be defined by x ( t ) = C ( t ) , y ( t ) = S ( t ) , t 0 . Then the set { x ( t ) , y ( t ) } is called Cornu’s spiral: it is the projection of the corkscrew on the { x , y } -plane. …
See accompanying text
Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by x = C ( t ) , y = S ( t ) , t [ 0 , ) . Magnify
5: 28.2 Definitions and Basic Properties
§28.2(i) Mathieu’s Equation
§28.2(iii) Floquet’s Theorem and the Characteristic Exponents
This is the characteristic equation of Mathieu’s equation (28.2.1). …
§28.2(iv) Floquet Solutions
6: 7.2 Definitions
§7.2(ii) Dawson’s Integral
7.2.5 F ( z ) = e z 2 0 z e t 2 d t .
7.2.8 S ( z ) = 0 z sin ( 1 2 π t 2 ) d t ,
( z ) , C ( z ) , and S ( z ) are entire functions of z , as are f ( z ) and g ( z ) in the next subsection. …
lim x S ( x ) = 1 2 .
7: 28.20 Definitions and Basic Properties
§28.20(i) Modified Mathieu’s Equation
When z is replaced by ± i z , (28.2.1) becomes the modified Mathieu’s equation:
28.20.1 w ′′ ( a 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 1 ) w ′′ + ζ w + ( 4 q ζ 2 2 q a ) w = 0 , ζ = cosh z .
For s , …
8: 23.7 Quarter Periods
§23.7 Quarter Periods
23.7.1 ( 1 2 ω 1 ) = e 1 + ( e 1 e 3 ) ( e 1 e 2 ) = e 1 + ω 1 2 ( K ( k ) ) 2 k ,
23.7.2 ( 1 2 ω 2 ) = e 2 i ( e 1 e 2 ) ( e 2 e 3 ) = e 2 i ω 1 2 ( K ( k ) ) 2 k k ,
23.7.3 ( 1 2 ω 3 ) = e 3 ( e 1 e 3 ) ( e 2 e 3 ) = e 3 ω 1 2 ( K ( k ) ) 2 k ,
9: 22.16 Related Functions
§22.16(ii) Jacobi’s Epsilon Function
Integral Representations
§22.16(iii) Jacobi’s Zeta Function
Definition
Properties
10: 23.4 Graphics
§23.4(i) Real Variables
Line graphs of the Weierstrass functions ( x ) , ζ ( x ) , and σ ( x ) , illustrating the lemniscatic and equianharmonic cases. …
See accompanying text
Figure 23.4.6: σ ( x ; 0 , g 3 ) for 5 x 5 , g 3 = 0. … Magnify
§23.4(ii) Complex Variables
Surfaces for the Weierstrass functions ( z ) , ζ ( z ) , and σ ( z ) . …