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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: 28.26 Asymptotic Approximations for Large q
§28.26(i) Goldsteins Expansions
Then as h + with fixed z in z > 0 and fixed s = 2 m + 1 ,
28.26.4 Fc m ( z , h ) 1 + s 8 h cosh 2 z + 1 2 11 h 2 ( s 4 + 86 s 2 + 105 cosh 4 z s 4 + 22 s 2 + 57 cosh 2 z ) + 1 2 14 h 3 ( s 5 + 14 s 3 + 33 s cosh 2 z 2 s 5 + 124 s 3 + 1122 s cosh 4 z + 3 s 5 + 290 s 3 + 1627 s cosh 6 z ) + ,
28.26.5 Gc m ( z , h ) sinh z cosh 2 z ( s 2 + 3 2 5 h + 1 2 9 h 2 ( s 3 + 3 s + 4 s 3 + 44 s cosh 2 z ) + 1 2 14 h 3 ( 5 s 4 + 34 s 2 + 9 s 6 47 s 4 + 667 s 2 + 2835 12 cosh 2 z + s 6 + 505 s 4 + 12139 s 2 + 10395 12 cosh 4 z ) ) + .
For additional terms see Goldstein (1927). …
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: 28.8 Asymptotic Expansions for Large q
Denote h = q and s = 2 m + 1 . …
§28.8(iii) Goldsteins Expansions
Barrett’s Expansions
Dunster’s Approximations