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- Homework Statement
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False
The reasoning for answer:
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!docnet said:Homework Statement:: .
Relevant Equations:: .
View attachment 294651
False
The reasoning for answer:
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.
really? that is surprising to me.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.
the zeroes of ##f## are given by $$|z|^2=2zRe(z)$$S.G. Janssens said:What can you say about the zeros of ##f## in the complex plane?
okay.. that makes sensePeroK said:Take as an example: $$f(z) = z = Re(z) +iIm(z)$$ is the sum of two non-analytic functions!
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.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.
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.docnet said:okay.. that makes sense
Oh gosh.. I forgot that the absolute value function has a formal definition ##|z|=\sqrt{z\bar{z}}##. I'm so sorry @PeroKdocnet 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 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.
You would write ##z=x+iy## (so ##\text{Re}(z)=x## and ##|z|^2=x^2+y^2##).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.
Would you believe it!docnet said:so ##f=|z|^2-2zRe(z)## satisfies the Cauchy Riemann equations and so it's entire.
You'd certainly bash out a PhD thesis quickly enough. Whether it would be defensible is another matter!docnet said:There goes my dreams for a Ph.D in maths, smashed to granules.
An entire function is a complex-valued function that is defined and analytic in the entire complex plane.
|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.
One example of an entire function is f(z) = e^z, where e is the base of the natural logarithm.
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.
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.