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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: 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)). …
9: 3.8 Nonlinear Equations
§3.8(ii) Newton’s Rule
Steffensen’s Method
Example. Wilkinsons Polynomial
10: 3.2 Linear Algebra
In the case that the orthogonality condition is replaced by 𝐒 -orthogonality, that is, 𝐯 j T 𝐒 𝐯 k = δ j , k , j , k = 1 , 2 , , n , for some positive definite matrix 𝐒 with Cholesky decomposition 𝐒 = 𝐋 T 𝐋 , then the details change as follows. Start with 𝐯 0 = 𝟎 , vector 𝐯 1 such that 𝐯 1 T 𝐒 𝐯 1 = 1 , α 1 = 𝐯 1 T 𝐀 𝐯 1 , β 1 = 0 . …
𝐮 = ( 𝐀 α j 𝐒 ) 𝐯 j β j 𝐒 𝐯 j 1 ,
Many methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).