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.

A Functional Equation Arising from Ivory's Theorem in Geometry

1975 ◽  
Vol 18 (4) ◽  
pp. 507-516
Author(s):  
Hiroshi Haruki
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Complex Variable ◽  
Complex Variables ◽  
Image Position

In previous papers (see [1, 2, 3, 4]), we solved the following functional equation:1wheref=f(z) is an entire function of a complex variable z and x, y are complex variables.

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.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

scholarly journals Dirichlet and Poincaré series

1985 ◽  
Vol 27 ◽  
pp. 39-56 ◽  
Author(s):  
A. Good
Keyword(s):  
Functional Equation ◽  
Number Theory ◽  
Algebraic Geometry ◽  
Entire Function ◽  
Cusp Form ◽  
Modular Forms ◽  
Modular Group ◽  
Positive Proportion ◽  
Fundamental Discriminant ◽  
Image Position

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet seriesdefines an entire function of s and satisfies the functional equationThe most penetrating statements that have been made on T(n) and LΔ(s)are:Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

Linear Combinations of Univalent Functions with Complex Coefficients

1971 ◽  
Vol 23 (4) ◽  
pp. 712-717 ◽  
Author(s):  
Robert K. Stump
Keyword(s):  
Linear Combination ◽  
Convex Domain ◽  
Analytic Functions ◽  
Univalent Functions ◽  
Linear Combinations ◽  
Radius Of Starlikeness ◽  
Complex Constant ◽  
Complex Functions ◽  
Image Position ◽  
Complex Coefficients

Let U be the class of all normalized analytic functionswhere z ∈ E = {z : |z| < 1} and ƒ is univalent in E. Let K denote the sub-class of U consisting of those members that map E onto a convex domain. MacGregor [2] showed that if ƒ1 ∈ K and ƒ2 ∈ K and if1then F ∉ K when λ is real and 0 < λ < 1, and the radius of univalency and starlikeness for F is .In this paper, we examine the expression (1) when ƒ1 ∈ K, ƒ2 ∈ K and λ is a complex constant and find the radius of starlikeness for such a linear combination of complex functions with complex coefficients.

Tauberian Theorems for Borel-Type Methods of Summability

1974 ◽  
Vol 17 (2) ◽  
pp. 167-173 ◽  
Author(s):  
D. Borwein ◽  
E. Smet
Keyword(s):  
Complex Variable ◽  
Negative Integer ◽  
Complex Numbers ◽  
Tauberian Theorems ◽  
Arbitrary Complex ◽  
Principal Value ◽  
Image Position ◽  
Borel Type

Suppose throughout that s, an (n=0,1, 2,…) are arbitrary complex numbers, that α>0 and β is real and that N is a non-negative integer such that αN+β≧l. Letwhere z=x+iy is a complex variable and the power zr is assumed to have its principal value.

On Some Relations Between Partial and Ordinary Differential Equations

1958 ◽  
Vol 10 ◽  
pp. 183-190 ◽  
Author(s):  
Erwin Kreyszig
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Operators ◽  
Basic Tool ◽  
Partial Differential ◽  
Image Position

The theory of solutions of partial differential equations (1.1) with analytic coefficients can be based upon the theory of analytic functions of a complex variable; the basic tool in this approach is integral operators which map the set of solutions of (1.1) onto the algebra of analytic functions. For certain classes of operators this mapping which is first defined in the small, can be continued to the large, cf. Bergman (3).

On the approximation of analytic functions in a strip

1985 ◽  
Vol 97 (3) ◽  
pp. 381-384 ◽  
Author(s):  
Dieter Klusch
Keyword(s):  
Entire Function ◽  
Real Axis ◽  
Exponential Type ◽  
Uniform Approximation ◽  
Analytic Functions ◽  
Trigonometric Polynomials ◽  
The Real ◽  
Image Position ◽  
Class Of Functions

1. Letand denote by Aδ the class of functions f analytic in the strip Sδ = {z = x + iy| |y| < δ}, real on the real axis, and satisfying |Ref(z)| ≤ 1,z∊Sδ. Then N.I. Achieser ([1], pp. 214–219; [8], pp. 137–8, 149) proved that each f∊Aδ can be uniformly approximated on the whole real axis by an entire function fc of exponential type at most c with an errorwhere ∥·∥∞ is the sup norm on ℝ. Furthermore ([7], pp. 196–201), if f∊Aδ is 2π-periodic, then the uniform approximation Ẽn (Aδ) of the class Aδ by trigonometric polynomials of degree at most n is given by

The Second Mean Values of Entire Functions

1966 ◽  
Vol 18 ◽  
pp. 1113-1120
Author(s):  
Q. I. Rahman
Keyword(s):  
Entire Function ◽  
Convex Function ◽  
Dirichlet Series ◽  
Complex Variable ◽  
Entire Functions ◽  
Mean Values ◽  
Image Position

Let f(z) be an entire function of the complex variable z = x + iy defined by the everywhere absolutely convergent Dirichlet series1.1Ifthen log m(x,f) is an increasing convex function of x (2), andis called the Ritt order of f(z).

A Functional Equation for the Cosine

1968 ◽  
Vol 11 (3) ◽  
pp. 495-498 ◽  
Author(s):  
PL Kannappan
Keyword(s):  
Functional Equation ◽  
Measurable Function ◽  
Trivial Solution ◽  
Real Variable ◽  
Real Constant ◽  
Real Numbers ◽  
Measurable Solution ◽  
Image Position ◽  
Complex Valued ◽  
Cosine Functions

It is known [3], [5] that, the complex-valued solutions of(B)(apart from the trivial solution f(x)≡0) are of the form(C)(D)In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],(E)In this paper, the functional equation(P)where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.

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