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11: 23.15 Definitions
A modular function f ( τ ) is a function of τ that is meromorphic in the half-plane τ > 0 , and has the property that for all 𝒜 SL ( 2 , ) , or for all 𝒜 belonging to a subgroup of SL ( 2 , ) , …
12: 25.15 Dirichlet L -functions
The notation L ( s , χ ) was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series …
13: 2.5 Mellin Transform Methods
If f ( 1 - z ) and h ( z ) can be continued analytically to meromorphic functions in a left half-plane, and if the contour z = c can be translated to z = d with d < c , then …Similarly, if f ( 1 - z ) and h ( z ) can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for I ( x ) for large values of x . … Furthermore, f 1 ( z ) can be continued analytically to a meromorphic function on the entire z -plane, whose singularities are simple poles at - α s , s = 0 , 1 , 2 , , with principal part … Similarly, if κ = 0 in (2.5.18), then h 2 ( z ) can be continued analytically to a meromorphic function on the entire z -plane with simple poles at β s , s = 0 , 1 , 2 , , with principal part … Furthermore, each G j k ( z ) has an analytic or meromorphic extension to a half-plane containing D j k . …
14: 8.2 Definitions and Basic Properties
When z 0 , Γ ( a , z ) is an entire function of a , and γ ( a , z ) is meromorphic with simple poles at a = - n , n = 0 , 1 , 2 , , with residue ( - 1 ) n / n ! . …
15: 23.3 Differential Equations
As functions of g 2 and g 3 , ( z ; g 2 , g 3 ) and ζ ( z ; g 2 , g 3 ) are meromorphic and σ ( z ; g 2 , g 3 ) is entire. …
16: 25.16 Mathematical Applications
H ( s ) is analytic for s > 1 , and can be extended meromorphically into the half-plane s > - 2 k for every positive integer k by use of the relations …
17: 29.12 Definitions
In consequence they are doubly-periodic meromorphic functions of z . …
18: 8.21 Generalized Sine and Cosine Integrals
Furthermore, si ( a , z ) and ci ( a , z ) are entire functions of a , and Si ( a , z ) and Ci ( a , z ) are meromorphic functions of a with simple poles at a = - 1 , - 3 , - 5 , and a = 0 , - 2 , - 4 , , respectively. …
19: 13.2 Definitions and Basic Properties
M ( a , b , z ) is entire in z and a , and is a meromorphic function of b . …
20: 25.11 Hurwitz Zeta Function
ζ ( s , a ) has a meromorphic continuation in the s -plane, its only singularity in being a simple pole at s = 1 with residue 1 . …