scholarly journals A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions

Nature ◽  
1920 ◽  
Vol 106 (2669) ◽  
pp. 531-531
Keyword(s):  
General Theory ◽  
Analytic Functions ◽  
Modern Analysis ◽  
Transcendental Functions

scholarly journals ALGEBRAIC VALUES OF CERTAIN ANALYTIC FUNCTIONS DEFINED BY A CANONICAL PRODUCT

2019 ◽  
Vol 101 (3) ◽  
pp. 415-425
Author(s):  
TABOKA P. CHALEBGWA
Keyword(s):  
Analytic Functions ◽  
Partial Answer ◽  
Canonical Product ◽  
Transcendental Functions

We give a partial answer to a question attributed to Chris Miller on algebraic values of certain transcendental functions of order less than one. We obtain $C(\log H)^{\unicode[STIX]{x1D702}}$ bounds for the number of algebraic points of height at most $H$ on certain subsets of the graphs of such functions. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ depend on data associated with the functions and can be effectively computed from them.

On the representation of analytic functions by infinite series

1953 ◽  
Vol 245 (900) ◽  
pp. 429-468 ◽  
Keyword(s):  
General Theory ◽  
Infinite Series ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Methods ◽  
Basic Series ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Product Sets

Given a set of functions {p k {z)}, necessary and sufficient conditions are known under which the basic series ∑ (k=0) ∞ II k f (0)p k (z) will represent all functions f ( z ) in certain classes. The various cases are included in a general theory given in part II. Questions of uniqueness are discussed, and an attempt is made to initiate a theory of representation by series of the form ∑ (k=0) ∞ α k p k (z) which are not necessarily basic. Topological methods are used, and part I is devoted largely to preliminaries. In part III is discussed the relationship between given sets and various associated sets such as the inverse and product sets.

scholarly journals ON A PROBLEM POSED BY MAHLER

2015 ◽  
Vol 100 (1) ◽  
pp. 86-107 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOHANNES SCHLEISCHITZ
Keyword(s):  
Rational Number ◽  
Analytic Functions ◽  
Rational Functions ◽  
Liouville Numbers ◽  
Transcendental Functions

Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that they can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to$f(z)=z^{q}$, where$q$is a rational number.

scholarly journals XIV. On the convergence of infinite series of analytic functions

1905 ◽  
Vol 204 (372-386) ◽  
pp. 481-497
Keyword(s):  
Analytic Function ◽  
General Theory ◽  
Infinite Series ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Point Of View ◽  
General Point ◽  
Associated Functions ◽  
Linear Differential ◽  
Series Of Functions

In the first section of the following work an attempt is made to deal with the convergence of infinite series of functions defined by linear differential equations of the second order from the most general point of view. Functions of Lamé Bessel and Legendre are considered as examples. In the second section the results obtained are applied to the expansion of an arbitrary uniform analytic function of an arbitrary uniform analytic function of z in a series of hypergeometric functions, and the expansion is shown to be valid if the function is regular within a certain ellipse in the z -plane. An expansion in a series of Legendre’s associated functions is deduced by a transformation. The method has been applied by the writer to other cases, but the foregoing offer adequate illustration of the general theory.

scholarly journals Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

2011 ◽  
Vol 147 (5) ◽  
pp. 1581-1607 ◽  
Author(s):  
Heiko Gimperlein ◽  
Bernhard Krötz ◽  
Henrik Schlichtkrull
Keyword(s):  
General Theory ◽  
Representation Theory ◽  
Lie Groups ◽  
Lie Group ◽  
Analytic Functions ◽  
General Framework ◽  
Analytic Representation ◽  
Reductive Groups ◽  
Topological Properties ◽  
Rapid Decay

AbstractIn this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra 𝒜(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and 𝒜(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.

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