Is this complex function analytic?

[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

 

[Auto-Didact]

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.

 

[dyn]

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

 

[Auto-Didact]

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.

 

[dyn]

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

 

[Auto-Didact]

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:
$$\begin{align}
u_x &= v_y \\
u_y &= -v_x \\
x &= 0
\end{align}$$

 

[dyn]

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

 

[Auto-Didact]

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.

 

[Auto-Didact]

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.

 

[FactChecker]

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.

 

[dyn]

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 ?

 

[FactChecker]

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]

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 ?

 

[Auto-Didact]

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

 

[vela]

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.