Table of Contents
- 1 Is an infinitely differentiable function analytic?
- 2 Which function is nowhere analytic?
- 3 What is continuously differentiable function?
- 4 Is Z Bar analytic?
- 5 Is the conjugate function holomorphic?
- 6 Can a Fourier series be used to construct an infinitely differentiable function?
- 7 Is the function f analytic in ℝ?
- 8 Can this pathology occur with differentiable functions of complex variables?
Is an infinitely differentiable function analytic?
Analyticity and differentiability As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or C∞). It can be proved that any complex function differentiable (in the complex sense) in an open set is analytic.
Which function is nowhere analytic?
Using the definition of differentiability I found out that the f(z) is differentiable at zero. But since it is not differentiable in a neighbourhood of zero therefore it cannot be analytic at zero and hence is nowhere analytic.
Are all differentiable functions analytic?
A differentiable function is said to be analytic at a point if the function is differentiable at and its neighbourhood. More precisely, If is an open set, i.e every point in is an interior point, and is complex differentiable at every point in , then is said to be analytic on .
What is continuously differentiable function?
A function is said to be continuously differentiable if the derivative exists and is itself a continuous function. Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.
Is Z Bar analytic?
It is not analytic because it is notcomplex-differentiable. You can see this by testing the Cauchy-Riemann equations. …
Does smooth mean infinitely differentiable?
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth.
Is the conjugate function holomorphic?
Given a harmonic function u : Ω → R, a function v : Ω → R is said to be a conjugate harmonic function if f = u+iv is a holomorphic function.
Can a Fourier series be used to construct an infinitely differentiable function?
A more pathological example, of an infinitely differentiable function which is not analytic at any point can be constructed by means of a Fourier series as follows. Let A := { 2 n : n ∈ ℕ } be the set of all powers of 2, and define for all x ∈ ℝ
What is a non-analytic smooth function?
The non-analytic smooth function f ( x) considered in the article. defined for every real number x . The function f has continuous derivatives of all orders at every point x of the real line. The formula for these derivatives is
Is the function f analytic in ℝ?
Since the set of analyticity of a function is an open set, and since dyadic rationals are dense, we conclude that F is nowhere analytic in ℝ. For every sequence α 0, α 1, α 2, . . . of real or complex numbers, the following construction shows the existence of a smooth function F on the real line which has these numbers as derivatives at the origin.
Can this pathology occur with differentiable functions of complex variables?
This pathology cannot occur with differentiable functions of a complex variable rather than of a real variable.