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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
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
§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 ,
For s ,
M ν ( 1 ) ( z + s π i , h ) = e i s π ν M ν ( 1 ) ( z , h ) ,
7: 22.16 Related Functions
§22.16(ii) Jacobi’s Epsilon Function
Integral Representations
§22.16(iii) Jacobi’s Zeta Function
Definition
Properties
8: 19.2 Definitions
Let s 2 ( t ) be a cubic or quartic polynomial in t with simple zeros, and let r ( s , t ) be a rational function of s and t containing at least one odd power of s . …Because s 2 is a polynomial, we have …
§19.2(ii) Legendre’s Integrals
§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)). …
9: 28.8 Asymptotic Expansions for Large q
§28.8(iii) Goldstein’s Expansions
Barretts Expansions
Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). …
Dunster’s Approximations
10: Bibliography B
  • R. F. Barrett (1964) Tables of modified Struve functions of orders zero and unity.
  • W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
  • B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.
  • B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.
  • M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.