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special values of the variable

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11: 25.11 Hurwitz Zeta Function
25.11.11 ζ ( s , 1 2 ) = ( 2 s 1 ) ζ ( s ) , s 1 .
25.11.15 ζ ( s , k a ) = k s n = 0 k 1 ζ ( s , a + n k ) , s 1 , k = 1 , 2 , 3 , .
25.11.16 ζ ( 1 s , h k ) = 2 Γ ( s ) ( 2 π k ) s r = 1 k cos ( π s 2 2 π r h k ) ζ ( s , r k ) , s 0 , 1 ; h , k integers, 1 h k .
12: 14.5 Special Values
14.5.14 𝖰 ν 1 / 2 ( cos θ ) = ( π 2 sin θ ) 1 / 2 cos ( ( ν + 1 2 ) θ ) ν + 1 2 .
14.5.23 𝖰 1 2 ( cos θ ) = K ( cos ( 1 2 θ ) ) .
13: 28.1 Special Notation
§28.1 Special Notation
(For other notation see Notation for the Special Functions.)
m , n integers.
x , y real variables.
z = x + i y complex variable.
λ ν ( q ) .
14: 12.1 Special Notation
§12.1 Special Notation
(For other notation see Notation for the Special Functions.)
x , y real variables.
z complex variable.
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …
15: 5.15 Polygamma Functions
5.15.1 ψ ( z ) = k = 0 1 ( k + z ) 2 , z 0 , 1 , 2 , ,
16: 8.21 Generalized Sine and Cosine Integrals
§8.21(i) Definitions: General Values
With γ and Γ denoting here the general values of the incomplete gamma functions (§8.2(i)), we define …
§8.21(ii) Definitions: Principal Values
§8.21(v) Special Values
§8.21(viii) Asymptotic Expansions
17: 4.23 Inverse Trigonometric Functions
§4.23(i) General Definitions
§4.23(ii) Principal Values
Compare the principal value of the logarithm (§4.2(i)). … Throughout this subsection all quantities assume their principal values. …
§4.23(vii) Special Values and Interrelations
18: 24.1 Special Notation
§24.1 Special Notation
(For other notation see Notation for the Special Functions.)
j , k , , m , n integers, nonnegative unless stated otherwise.
t , x real or complex variables.
Unless otherwise noted, the formulas in this chapter hold for all values of the variables x and t , and for all nonnegative integers n . …
19: 8.4 Special Values
§8.4 Special Values
8.4.5 Γ ( 1 , z ) = e z ,
8.4.9 P ( n + 1 , z ) = 1 e z e n ( z ) ,
8.4.10 Q ( n + 1 , z ) = e z e n ( z ) ,
8.4.11 e n ( z ) = k = 0 n z k k ! .
20: 25.16 Mathematical Applications
25.16.10 H ( 2 a ) = 1 2 ζ ( 1 2 a ) = B 2 a 4 a , a = 1 , 2 , 3 , .
H ( s ) is the special case H ( s , 1 ) of the function …
25.16.13 n = 1 ( H n n ) 2 = 17 4 ζ ( 4 ) ,
25.16.14 r = 1 k = 1 r 1 r k ( r + k ) = 5 4 ζ ( 3 ) ,
25.16.15 r = 1 k = 1 r 1 r 2 ( r + k ) = 3 4 ζ ( 3 ) .