Can analytic functions be constant?

Can analytic functions be constant?

are both real valued, complex variabled, analytic functions thus must be constant and thus f is constant.

How do you show that a function is constant?

A function is a constant function if f(x)=c f ( x ) = c for all values of x and some constant c . The graph of the constant function y(x)=c y ( x ) = c is a horizontal line in the plane that passes through the point (0,c).

Is an analytic function with constant modulus is constant?

⇒ u x = u y = 0 ⇒ u is a constant function. ∴ f(z) is a constant function.

Which of the following is not a analytic function?

C.R. equation is not satisfied. So, f(z)=|z|2 is not analytic.

How do you prove that a constant function is continuous?

Let a ∈ R be a constant, and let f be a function defined on an open interval containing a. We say f is continuous at a if limx→a f(x) = f(a). This is roughly equivalent to saying that a function is continuous if its graph can be drawn without lifting the pen.

What is an analytic function with constant modulus?

That means that, if an analytic function sends an open subset of into a one-dimensional curve, its derivative has to have rank zero everywhere, which means that its zero everywhere, therefore the function is constant.

Which function is not analytic?

If a function is not continous or differentiable then it is not analytic. Also, if you split a function, f(z) into f(x+iy)=u(x,y)+iv(x,y) and, ux≠vy and/or uy≠−vx then the function is not analytic. These are known as the Cauchy-Riemann equations and if they are not satisfied then the function is not analytic.

How do you find the constant of an analytic function?

This method uses the fact that if f and f ¯ are both analytic then f is constant. If | f | = 0 then f is always zero. If c = | f | > 0 we have c 2 = f f ¯ then f ¯ = c 2 / f. Since f ≠ 0 it follows that f ¯ is analytic, and hence f is constant.

How do you prove a function is a constant function?

An analytic function f in a region Ω whose argument a r g ( f) is constant must be a constant function. First he proves that if R e ( f) is a constant function then f is constant. I’ve understood his proof for that.

How do you prove that a function is analytic?

Proof :- Write where, and . Under the assumption that is constant we get, for some real constant . That is, is analytic. But, by Cauchy Riemann equation we get, . Hence also, . Since, is connected we conclude that and are constant functions, so is . This completes the proof.

How do you prove that a nonconstant holomorphic function is constant?

You can also deduce this from the open mapping theorem: a nonconstant holomorphic function is an open map. If | f | is constant, then f ( C) is contained in the circle of radius | f |, which has empty interior. Hence f is constant. If f is analytic on all of C then f is constant by Liouville’s theorem.