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functions s(ϵ,ℓ;r),c(ϵ,ℓ;r)

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1: 7.2 Definitions
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. …
7.2.10 f ( z ) = ( 1 2 - S ( z ) ) cos ( 1 2 π z 2 ) - ( 1 2 - C ( z ) ) sin ( 1 2 π z 2 ) ,
7.2.11 g ( z ) = ( 1 2 - C ( z ) ) cos ( 1 2 π z 2 ) + ( 1 2 - S ( z ) ) sin ( 1 2 π z 2 ) .
2: 22.16 Related Functions
§22.16(ii) Jacobi’s Epsilon Function
Integral Representations
Relation to Theta Functions
§22.16(iii) Jacobi’s Zeta Function
Definition
3: 28.2 Definitions and Basic Properties
§28.2(ii) Basic Solutions w I , w II
§28.2(iv) Floquet Solutions
28.2.18 w ( z ) = n = - c 2 n e i ( ν + 2 n ) z
See accompanying text
Figure 28.2.1: Eigenvalues a n ( q ) , b n ( q ) of Mathieu’s equation as functions of q for 0 q 10 , n = 0 , 1 , 2 , 3 , 4 ( a s), n = 1 , 2 , 3 , 4 ( b s). Magnify
§28.2(vi) Eigenfunctions
4: 28.20 Definitions and Basic Properties
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 .
28.20.6 Fe n ( z , q ) = i fe n ( ± i z , q ) , n = 0 , 1 , ,
28.20.7 Ge n ( z , q ) = ge n ( ± i z , q ) , n = 1 , 2 , .
5: 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 . …
19.2.1 r ( s , t ) d t
19.2.2 r ( s , t ) = ( p 1 + p 2 s ) ( p 3 - p 4 s ) s ( p 3 + p 4 s ) ( p 3 - p 4 s ) s = ρ s + σ ,
19.2.17 R C ( x , y ) = 1 2 0 d t t + x ( t + y ) ,
19.2.21 R C ( x , y ) = 0 1 ( v 2 x + ( 1 - v 2 ) y ) - 1 / 2 d v ,
6: 25.1 Special Notation
The main function treated in this chapter is the Riemann zeta function ζ ( s ) . … The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
7: 31.6 Path-Multiplicative Solutions
8: 4.44 Other Applications
The Einstein functions and Planck’s radiation function are elementary combinations of exponentials, or exponentials and logarithms. …
9: 31.4 Solutions Analytic at Two Singularities: Heun Functions
To emphasize this property this set of functions is denoted by The set q m depends on the choice of s 1 and s 2 . …
10: 31.1 Special Notation
The main functions treated in this chapter are H ( a , q ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ( a , q m ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ν ( a , q m ; α , β , γ , δ ; z ) , and the polynomial Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) . …Sometimes the parameters are suppressed.