Which function is not analytic?

The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.

Table of Contents

How do you show that a function 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.

Is 0 an analytic function?

1 Page 2 The zero function is the only analytic function that has a zero of infinite order.

Is the function f z )= z is analytic?

f(z)=|z| is not analytic.

Is Cos z analytic?

So sin z is not analytic anywhere. Similarly cos z = cosxcosh y + isinxsinhy = u + iv, and the Cauchy-Riemann equations hold when z = nπ for n ∈ Z. Thus cosz is not analytic anywhere, for the same reason as above.

Is Sinx analytic?

The sin(x) function is both differentiable and analytic everywhere in the complex plane.

Is log z analytic?

Answer: The function Log(z) is analytic except when z is a negative real number or 0.

Is z analytic?

z* Is Not Analytic It is interesting to note that f (z) = z* is continuous, thus providing an example of a function that is everywhere continuous but nowhere differentiable in the complex plane.

Is z 2 analytic?

(a) z = x + iy, |z|2 = x2 + y2, u = x2, v = y2 ux = 2x = vy = 2y Hence not analytic. The partial derivatives are continuous and hence the function is ana- lytic.

Is f z z * analytic?

z* Is Not Analytic The Cauchy-Riemann conditions are not satisfied for any values of x or y and f (z) = z* is nowhere an analytic function of z. It is interesting to note that f (z) = z* is continuous, thus providing an example of a function that is everywhere continuous but nowhere differentiable in the complex plane.

Is SINZ analytic?

Is Tanz analytic?

Most of these can be proved using the definitions involving exponentials. For example, tanz and sec z are analytic everywhere except at the zeroes of cos z.

Is Cos an analytic function?

Similarly cos z = cosxcosh y + isinxsinhy = u + iv, and the Cauchy-Riemann equations hold when z = nπ for n ∈ Z. Thus cosz is not analytic anywhere, for the same reason as above.

Is sin z z analytic?

TRUE. Both sin(z) and f are analytic in D(0, 1) and agree on a sequence with a limit point in D(0, 1), so by the identity theorem they must be the same. (b) There exists an f(z) analytic in D(0, 1) and a sequence zn → 0 contained in D(0, 1) such that f(zn) = n for all n.

Is ZZ * analytic?

it’s not analytic. FYI these are some good properties to remember for exams.

Is Tan z analytic?

For example, tanz and sec z are analytic everywhere except at the zeroes of cos z.

Is Cos z analytic or not?

Is tan analytic?

There are a whole bunch of typical trigonometric identities which are valid for complex trig functions. Most of these can be proved using the definitions involving exponentials. For example, tanz and sec z are analytic everywhere except at the zeroes of cos z.

Is sin z analytical?

To show sinz is analytic. Hence the cauchy-riemann equations are satisfied. Thus sinz is analytic.

Is f z )= z3 analytic?

1) Show that f(z) = z3 is analytic. exists and continuous. Hence the given function f(z) is analytic.

Is z3 analytic?

Is cos2z analytic?

∂u/∂y = cos(x) sinh(y) and ∂v/∂x = -cos(x) sinh(y) = – ∂u/∂y for all (x, y). It follows that w = cos(z) is analytic everywhere in the plane.

Is SINZ an entire function?

sinz is thus, as you know, an entire function, defined and finite for all z∈C. Liouville’s theorem, that the only entire function that is bounded as z→∞ (bounded and analytic on the extended complex plane C⋃{∞}) is a constant, then gives you the rest.

Why is SINZ not analytic?