On a cosine functional equation for operators on the algebra of analytic functions in a domain

2001 ◽  
Vol 61 (3) ◽  
pp. 233-238 ◽  
Author(s):  
PL. Kannappan ◽  
N. R. Nandakumar
Keyword(s):  
Functional Equation ◽  
Analytic Functions ◽  
Cosine Functional Equation

XXIV.—The Solution of a Functional Equation

1952 ◽  
Vol 63 (4) ◽  
pp. 336-345
Author(s):  
A. H. Read
Keyword(s):  
Functional Equation ◽  
Recurrence Relation ◽  
Unknown Function ◽  
Analytic Functions ◽  
Analytic Solutions ◽  
Sequence Of Functions

SynopsisAnalytic solutions of the functional equation f[z, φ{g(z)}] = φ(z), in which f(z, w) and g(z) are given analytic functions and φ(z) is the unknown function, are investigated in the neighbourhood of points ζ such that g(ζ) = ζ. Conditions are established under which each solution φ(z) may be given as the limit of a sequence of functions φn(z), defined by the recurrence relation φn+1(Z) = ƒ[z, φn{g(z)}], the function φn(z) being to a large extent arbitrary.

A Cosine Functional Equation with Restricted Argument

1974 ◽  
Vol 17 (4) ◽  
pp. 505-509 ◽  
Author(s):  
L. B. Etigson
Keyword(s):  
Functional Equation ◽  
The Other ◽  
Single Element ◽  
Discrete Subset ◽  
Cosine Functional Equation ◽  
Cosine Functions

We name a functional equation with restricted argument one in which at least one of the variables is restricted to a certain discrete subset of the domain of the other variable(s). In particular, the subset may consist of a single element.The purpose of this paper is to present a functional equation satisfied only by cosine functions.

scholarly journals On a functional equation for the exponential function of a complex variable

1971 ◽  
Vol 12 (1) ◽  
pp. 31-34 ◽  
Author(s):  
Hiroshi Haruki
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Exponential Function ◽  
Complex Variable ◽  
Analytic Functions ◽  
Real Constant ◽  
Complex Constant ◽  
Arbitrary Real Constant ◽  
Arbitrary Complex ◽  
Image Position

The following result is well known in the theory of analytic functions; see [1].Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equationwhere z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.

scholarly journals The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

2008 ◽  
Vol 103 (1) ◽  
pp. 11 ◽  
Author(s):  
Christian Berg ◽  
Antonio j. Durán
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Meromorphic Function ◽  
Convex Function ◽  
Rational Function ◽  
Complex Plane ◽  
Linear Transformation ◽  
Gamma Function ◽  
Analytic Functions ◽  
Mellin Transform

We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: $T((a_n))_n=1/(a_0+\cdots +a_n)$. We determine the corresponding measure $\mu$, which has an increasing and convex density on $\mathopen]0,1\mathclose[$, and we study some analytic functions related to it. The Mellin transform $F$ of $\mu$ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on $\mathopen]-1,\infty\mathclose[$ satisfying $F(0)=1$ and the functional equation $1/F(s)=1/F(s+1)-F(s+1)$, $s>-1$.

scholarly journals An application of Nevanlinna-pólya theorem to a cosine functional equation

1970 ◽  
Vol 11 (3) ◽  
pp. 325-328 ◽  
Author(s):  
Hiroshi Haruki
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Complex Variable ◽  
Complex Variables ◽  
Cosine Functional Equation ◽  
Image Position

We consider the cosine functional equation (see [1, 2, 3]) , where f(z) is an entire function of a complex variable z and x, y are complex variables.

On an exponential-cosine functional equation

1988 ◽  
Vol 19 (4) ◽  
pp. 287-297 ◽  
Author(s):  
J. C. Parnami ◽  
H. Singh ◽  
H. L. Vasudeva
Keyword(s):  
Functional Equation ◽  
Cosine Functional Equation
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