# Is this complex function analytic?

• dyn
In the first case, the function is differentiable at every point on the circle. In the second case, the function is differentiable only at the point ##z=0##.
dyn
Homework Statement
Is f = u(x,y) + iv(x,y) an analytic function of z=x+iy where ## u(x,y) = x^3 - 3x(y^2) ## and ##v(x,y) = -y^3-3(x^2)y ##
Relevant Equations
The Cauchy-Riemann equations ,
## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ##

## u_y = -6xy## and ## v_x = -6xy##

To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x##

Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0

However the answer is that the function is not analytic ; I don't understand that. Any help would be appreciated

dyn said:
Problem Statement: Is f = u(x,y) + iv(x,y) an analytic function of z=x+iy where ## u(x,y) = x^3 - 3x(y^2) ## and ##v(x,y) = -y^3-3(x^2)y ##
Relevant Equations: The Cauchy-Riemann equations ,

## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ##

## u_y = -6xy## and ## v_x = -6xy##

To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x##

A function is (complex) analytic if and only if:
\begin{align} u_x &= v_y \\ u_y &= -v_x \end{align}
Just plug in and check that:
\begin{align} (1):& \quad 3x^2 -3y^2 \neq -3y^2-3x^2 \notag \\ (2):& \quad -6xy \neq -(-6xy) \notag \end{align}
Therefore the function is not analytic. Moreover, being analytic on some line which is a subset of the functions domain instead of on the functions entire domain is irrelevant.

I have seen examples where a function such as ##f(z) = x(y^2-1)-ix^2y ## which is differentiable(ie. analytic) only on the circle ##x^2+y^2 =1##
I also think I remember functions only being analytic at a certain point.
So , I don't understand why the function in #1 is not analytic on the line x=0

It is analytic on that line but that isn't what the original question was asking, therefore you are going off on an irrelevant tangent.

The question says use the C-R conditions to test whether the complex function f is an analytic function of z. The C-R equations say that f is analytic on the line x=0

But it isn't analytic on the rest of the domain, i.e. it isn't analytic for any value of ##x## and ##y##.

You aren't really using the Cauchy-Riemann equations, but instead a conditional version which is numeric instead of algebraic:
\begin{align} u_x &= v_y \\ u_y &= -v_x \\ x &= 0 \end{align}

But it is analytic for x=0 and any value of y , so the C-R conditions have shown that f is analytic on that particular line

I already agreed with that. What is your point? Is the function analytic on the line ##x=0##? Yes. Is the function analytic in general? No. Nothing more needs to be said.

Actually, if we try to approach the second Cauchy-Riemann equation ##u_y=-v_x## using a limit of ##x## to ##0## instead of ##0## itself, the function is clearly not analytic, since for any ## \lim_{x > 0}## we have that ##u_y## and ##-v_x## will always have opposite signs, meaning that they are complex conjugates and that the function is therefore by definition non-holomorphic i.e. not analytic.

Unless there is some indication, I think you are safe to assume that they are asking if the function is analytic in an open set. I have never seen any use for knowing if it is analytic at a single isolated point or a line with no surrounding open set.

Auto-Didact
I will try to explain my confusion more clearly using the following 2 examples.

Ex.1 By deriving the C-R equations for the function ## f(z) = x(y^2-1) - ix^2y ## infer the set of points (x,y) in the Argand diagram for which the function is differentiable.

Ex. 2 Use the C-R conditions to test if f=u(x,y)+iv(x,y) where ##u(x,y) = x^3 - 3x^2y## and ##v(x,y) = -y^3 - 3x^2y ## is an analytic function of z=x+iy

The answer for Ex.1 is that the function is differentiable on the circle ##x^2+y^2 = 1##

The answer for Ex.2 is that the 1st C-R condition is not satisfied as ##u_x = 3x^2-3y^2## and ## v_y = -3y^2- 3x^2##

My argument is that both C-R conditions are satisfied in Ex.2 for x=0 giving a straight vertical line through x=0.
I don't understand why a curve(circle) is a valid answer to Ex.1 but a curve(straight line) is not a valid answer to Ex.2 ?

It is not the difference between a curve versus a straight line that is important. The only explanation that I see is that the first example is phrased differently: "differentiable" versus "analytic"; "set of points" versus ??;
The language of the first example might imply a different type of answer is appropriate. The language of the second example would usually imply that the function is not analytic. I am only used to seeing the term "analytic" when it is applied to its properties in a region (an open set). So I would not consider the function to be analytic only on the line x=0.

dyn and Klystron
Does that mean a function can be differentiable on a line/curve or even at a point but if it is only differentiable on that line/curve/point it is not analytic ?

Yes, analyticity is a much stronger condition than differentiability, i.e. a function can be differentiable yet not be analytic but not vice versa.

dyn
dyn said:
Does that mean a function can be differentiable on a line/curve or even at a point but if it is only differentiable on that line/curve/point it is not analytic ?
According to Arfken and Weber, a complex function is differentiable at a point ##z=z_0## iff the Cauchy-Riemann conditions hold at that point. The function is analytic at ##z=z_0## if it's differentiable at that point and in some small region around ##z_0##. These definitions are consistent with the two examples you provided.

dyn and FactChecker

## 1. What is an analytic function?

An analytic function is a complex-valued function that can be represented by a convergent power series in a neighborhood of each point in its domain. In simpler terms, it is a function that is differentiable at every point in its domain.

## 2. How can I determine if a function is analytic?

To determine if a function is analytic, you can check if it satisfies the Cauchy-Riemann equations, which are a set of necessary and sufficient conditions for a function to be analytic. You can also check if the function can be represented by a power series or if it has a continuous derivative in its entire domain.

## 3. Is every complex function analytic?

No, not every complex function is analytic. Some functions may have singularities or discontinuities in their domain, which would make them non-analytic. Additionally, some functions may not satisfy the Cauchy-Riemann equations and therefore cannot be represented by a power series.

## 4. Can a function be analytic at some points but not others?

Yes, it is possible for a function to be analytic at some points in its domain but not at others. This would happen if the function has singularities or discontinuities at certain points, but is analytic everywhere else.

## 5. Why is it important to determine if a function is analytic?

Analytic functions have many important properties and are used in various areas of mathematics and science. They allow for easier calculations and have applications in fields such as physics, engineering, and economics. Additionally, the study of analytic functions is fundamental in complex analysis, which is a branch of mathematics that deals with functions of complex variables.

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