# True or false? |z|^2 is an entire function

• docnet
In summary, it was discussed that the absolute value function is not analytic wherever its argument equals zero. It was also mentioned that ##f## is not analytic at ##z=0##, therefore it is not entire. However, it was pointed out that in general, if ##f(z) = f_1(z) + f_2(z)##, then even if ##f_1## and ##f_2## are not analytic, their sum may be. It was suggested that the non-analytic behavior of the individual functions may cancel each other out. It was then noted that the absolute value function has a formal definition of ##|z|=\sqrt{z\bar{z}}## and that it is real and

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Homework Statement
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Relevant Equations
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False

The absolute value function is is not analytic wherever its argument equals zero. ##f## is not analytic at ##z=0## so it is not entire.

docnet said:
Homework Statement:: .
Relevant Equations:: .

View attachment 294651

False

The absolute value function is is not analytic wherever its argument equals zero. ##f## is not analytic at ##z=0## so it is not entire.
That's a fairly significant logical fallacy!

In general, if ##f(z) = f_1(z) + f_2(z)##, then even if ##f_1## and ##f_2## are not analytic, their sum may be. Intuitively, the non-analytic behaviour of ##f_1## and ##f_2## may cancel each other out in some way.

docnet and S.G. Janssens
What can you say about the zeros of ##f## in the complex plane?

docnet
PeroK said:
That's a fairly significant logical fallacy!

In general, if ##f(z) = f_1(z) + f_2(z)##, then even if ##f_1## and ##f_2## are not analytic, their sum may be. Intuitively, the non-analytic behaviour of ##f_1## and ##f_2## may cancel each other out in some way.
really? that is surprising to me.

Take as an example: $$f(z) = z = Re(z) +iIm(z)$$ is the sum of two non-analytic functions!

docnet
S.G. Janssens said:
What can you say about the zeros of ##f## in the complex plane?
the zeroes of ##f## are given by $$|z|^2=2zRe(z)$$
the left size is real valued, so the right side has to be real valued. The equation ##x^2=2x^2## has no solutions other than ##x=0##, so the zero of ##f## is ##z=0##.

You have one thing a bit backwards: ##z=0## is actually the only point where ##|z|^2## is complex differentiable. You can verify this from the Cauchy Riemann equations.

And anyway, a function ##f## being (not) differentiable at a point doesn't mean the same is true for ##f+g.##

For your function, you can also write it in real and imaginary parts and see whether the Cauchy-Riemann equations hold.

Edit: Apologies, while I was writing this, several posts appeared that i didn't notice!

docnet
PeroK said:
Take as an example: $$f(z) = z = Re(z) +iIm(z)$$ is the sum of two non-analytic functions!
okay.. that makes sense

Infrared said:
You have one thing a bit backwards: ##z=0## is actually the only point where ##|z|^2## is complex differentiable. You can verify this from the Cauchy Riemann equations.

And anyway, a function ##f## being (not) differentiable at a point doesn't mean the same is true for ##f+g.##

For your function, you can also write it in real and imaginary parts and see whether the Cauchy-Riemann equations hold.
How do you tell the real and imaginary parts of ##z## in the absolute value function? this is what I wanted to do, compute the Cauchy Riemann equations, but the absolute value function confused me.

docnet said:
okay.. that makes sense
You know what I'm going to say ... you did the same thing again! You jumped at an easy, one-line proof, that you didn't seriously question. You must be able to fully justify each step you take.

Developing your maths skills means developing the ability to find the example I gave in post #5.

docnet
docnet said:
How do you tell the real and imaginary parts of ##z## in the absolute value function? this is what I wanted to do, compute the Cauchy Riemann equations, but the absolute value function confused me.
Oh gosh.. I forgot that the absolute value function has a formal definition ##|z|=\sqrt{z\bar{z}}##. I'm so sorry @PeroK

docnet said:
How do you tell the real and imaginary parts of ##z## in the absolute value function? this is what I wanted to do, compute the Cauchy Riemann equations, but the absolute value function confused me.
The absolute value is real, it has no imaginary part.

docnet
docnet said:
How do you tell the real and imaginary parts of in the value funciton? this is what I wanted to do, compute the Cauchy Riemann equations, but the absolute value funciton confused me.
You would write ##z=x+iy## (so ##\text{Re}(z)=x## and ##|z|^2=x^2+y^2##).

docnet and PeroK
$$f=|z|^2-2zRe(z)=x^2+y^2-2(x+iy)x$$
$$=(y^2-x^2)-i(2xy)$$
$$v_x=-2x=u_y$$
$$-v_y=-2y=u_x$$
so ##f=|z|^2-2zRe(z)## satisfies the Cauchy Riemann equations and so it's entire.

PeroK
docnet said:
so ##f=|z|^2-2zRe(z)## satisfies the Cauchy Riemann equations and so it's entire.
Would you believe it!

docnet
There goes my dreams for a Ph.D in maths, smashed to granules.

docnet said:
There goes my dreams for a Ph.D in maths, smashed to granules.
You'd certainly bash out a PhD thesis quickly enough. Whether it would be defensible is another matter!

docnet

## 1. What is an entire function?

An entire function is a complex-valued function that is defined and analytic in the entire complex plane.

## 2. What is the meaning of |z|^2 in the statement?

|z|^2 refers to the modulus or absolute value squared of the complex number z. In other words, it is the distance of z from the origin squared.

## 3. Can you give an example of an entire function?

One example of an entire function is f(z) = e^z, where e is the base of the natural logarithm.

## 4. How can you determine if a function is entire?

A function is entire if it is defined and analytic in the entire complex plane. This means that it must have a derivative at every point in the complex plane.

## 5. What is the significance of |z|^2 being an entire function?

If |z|^2 is an entire function, it means that it is defined and analytic in the entire complex plane. This also implies that it is a continuous and differentiable function in the complex plane, making it a useful tool in complex analysis and other areas of mathematics.