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1: 25.1 Special Notation
k , m , n nonnegative integers.
z = x + i y complex variable.
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 ) .
2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
For any partition κ , the zonal polynomial Z κ : 𝓢 is defined by the properties …
Normalization
Orthogonal Invariance
Summation
3: 23.13 Zeros
§23.13 Zeros
For information on the zeros of ( z ) see Eichler and Zagier (1982).
4: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
§25.11(i) Definition
The Riemann zeta function is a special case: …
See accompanying text
Figure 25.11.1: Hurwitz zeta function ζ ( x , a ) , a = 0. … Magnify
See accompanying text
Figure 25.11.2: Hurwitz zeta function ζ ( x , a ) , 19.5 x 10 , 0.02 a 1 . Magnify 3D Help
5: 10.58 Zeros
§10.58 Zeros
a n , m = j n + 1 2 , m ,
b n , m = y n + 1 2 , m ,
𝗃 n ( a n , m ) = π 2 j n + 1 2 , m J n + 1 2 ( j n + 1 2 , m ) ,
𝗒 n ( b n , m ) = π 2 y n + 1 2 , m Y n + 1 2 ( y n + 1 2 , m ) .
6: 9.9 Zeros
§9.9 Zeros
They are denoted by a k , a k , b k , b k , respectively, arranged in ascending order of absolute value for k = 1 , 2 , . For error bounds for the asymptotic expansions of a k , b k , a k , and b k see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …
Table 9.9.1: Zeros of Ai and Ai .
k a k Ai ( a k ) a k Ai ( a k )
Table 9.9.2: Real zeros of Bi and Bi .
k b k Bi ( b k ) b k Bi ( b k )
7: 32.5 Integral Equations
Let K ( z , ζ ) be the solution of
32.5.1 K ( z , ζ ) = k Ai ( z + ζ 2 ) + k 2 4 z z K ( z , s ) Ai ( s + t 2 ) Ai ( t + ζ 2 ) d s d t ,
where k is a real constant, and Ai ( z ) is defined in §9.2. …
32.5.2 w ( z ) = K ( z , z ) ,
32.5.3 w ( z ) k Ai ( z ) , z + .
8: 4.14 Definitions and Periodicity
4.14.7 cot z = cos z sin z = 1 tan z .
The functions sin z and cos z are entire. In the zeros of sin z are z = k π , k ; the zeros of cos z are z = ( k + 1 2 ) π , k . The functions tan z , csc z , sec z , and cot z are meromorphic, and the locations of their zeros and poles follow from (4.14.4) to (4.14.7). For k
9: 4.28 Definitions and Periodicity
4.28.2 cosh z = e z + e z 2 ,
4.28.9 cos ( i z ) = cosh z ,
4.28.12 sec ( i z ) = sech z ,
Periodicity and Zeros
The zeros of sinh z and cosh z are z = i k π and z = i ( k + 1 2 ) π , respectively, k .
10: 10.21 Zeros
§10.21 Zeros
j 0 , 1 = 0 ,
j 0 , m = j 1 , m 1 , m = 2 , 3 , .
j ν , 1 < j ν + 1 , 1 < j ν , 2 < j ν + 1 , 2 < j ν , 3 < ,
Next, z ( ζ ) is the inverse of the function ζ = ζ ( z ) defined by (10.20.3). …