scholarly journals On exponential representations of analytic functions in the upper half-plane with positive imaginary part

1956 ◽  
Vol 5 (1) ◽  
pp. 321-388 ◽  
Author(s):  
N. Aronszajn ◽  
W. F. Donoghue
Keyword(s):  
Half Plane ◽  
Imaginary Part ◽  
Analytic Functions ◽  
Positive Imaginary Part ◽  
Upper Half Plane

scholarly journals Differential Subordination Results for Analytic Functions in the Upper Half-Plane

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Huo Tang ◽  
M. K. Aouf ◽  
Guan-Tie Deng ◽  
Shu-Hai Li
Keyword(s):  
Unit Disk ◽  
Half Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Upper Half Plane ◽  
Admissible Functions

There are many articles in the literature dealing with differential subordination problems for analytic functions in the unit disk, and only a few articles deal with the above problems in the upper half-plane. In this paper, we aim to derive several differential subordination results for analytic functions in the upper half-plane by investigating certain suitable classes of admissible functions. Some useful consequences of our main results are also pointed out.

scholarly journals A discrete analogue of the Paley-Wiener theorem for bounded analytic functions in a half plane

1977 ◽  
Vol 23 (3) ◽  
pp. 376-378
Author(s):  
Doron Zeilberger
Keyword(s):  
Analytic Function ◽  
Half Plane ◽  
Analytic Functions ◽  
Complex Analysis ◽  
Discrete Analogue ◽  
Bounded Analytic Functions ◽  
Bounded Analytic Function ◽  
Wiener Theorem ◽  
Upper Half Plane

In this note we prove a discrete analogue to the following Paley–Weiner theorem: Let f be the restriction to the line of a bounded analytic function in the upper half plane; then the spectrum of f is contained in ([0, ∈). The discrete analogue of complex analysis is the theory of discrete analytic functions invented by Lelong-Ferrand (1944) and developed by Duffin (1956) and others.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

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