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1: 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
2: 31.2 Differential Equations
§31.2(i) Heun’s Equation
Jacobi’s Elliptic Form
Weierstrass’s Form
§31.2(v) Heun’s Equation Automorphisms
Composite Transformations
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: 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
5: 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 .
6: 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 , …
7: 22.16 Related Functions
§22.16(ii) Jacobi’s Epsilon Function
Integral Representations
§22.16(iii) Jacobi’s Zeta Function
Definition
Properties
8: 20.1 Special Notation
m , n integers.
S 1 / S 2 set of all elements of S 1 , modulo elements of S 2 . Thus two elements of S 1 / S 2 are equivalent if they are both in S 1 and their difference is in S 2 . (For an example see §20.12(ii).)
Jacobi’s original notation: Θ ( z | τ ) , Θ 1 ( z | τ ) , H ( z | τ ) , H 1 ( z | τ ) , respectively, for θ 4 ( u | τ ) , θ 3 ( u | τ ) , θ 1 ( u | τ ) , θ 2 ( u | τ ) , where u = z / θ 3 2 ( 0 | τ ) . … Nevilles notation: θ s ( z | τ ) , θ c ( z | τ ) , θ d ( z | τ ) , θ n ( z | τ ) , respectively, for θ 3 2 ( 0 | τ ) θ 1 ( u | τ ) / θ 1 ( 0 | τ ) , θ 2 ( u | τ ) / θ 2 ( 0 | τ ) , θ 3 ( u | τ ) / θ 3 ( 0 | τ ) , θ 4 ( u | τ ) / θ 4 ( 0 | τ ) , where again u = z / θ 3 2 ( 0 | τ ) . This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951). McKean and Moll’s notation: ϑ j ( z | τ ) = θ j ( π z | τ ) , j = 1 , 2 , 3 , 4 . …
9: 19.2 Definitions
Because s 2 is a polynomial, we have …
§19.2(ii) Legendre’s Integrals
Legendre’s complementary complete elliptic integrals are defined via …
§19.2(iii) Bulirsch’s Integrals
Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). …
10: 22.2 Definitions
Glaisher’s Notation
Let p , q , r be any three of the letters s , c , d , n . … s s ( z , k ) = 1 . The six functions containing the letter s in their two-letter name are odd in z ; the other six are even in z . In terms of Nevilles theta functions (§20.1) …