Complex Analysis: prove the function is entire

In summary, the conversation is about a student struggling with a homework question in their complex analysis course. They are trying to prove that a function is entire, but are unsure of the best approach. They consider expanding the function and using the CR-equations, but are also unsure about using the criteria of partial(f)/partial(zbar)=0. Eventually, they realize the correct method and reflect on their understanding of the conjugate of z equation.
  • #1
dh363
33
0
Hey guys, i just started a complex analysis course this semester and we just went over CR-equations, and various ways to show that a function is holomorphic. I'm a bit stuck on this one homework question where we have to prove the function is entire.

Homework Statement



so we have f(z)=(z^2-2)(e^-x)(e^-iy)

The Attempt at a Solution


First I tried expanding the e^-iy out to cos(-y)+isin(-y) and tried to multiply it out to get a u and a v. That required a lot of tedious algebraic manipulation for one of the earlier problems in the pset for this prof so then I tried looking for a more elegant solution. I turned (e^-x)(e^-iy) into e^(-z), and tried to show that the partial derivative with respect too the conjugate of z = 0, but I'm still not to clear on that whole method. Was wondering if anyone can point me along the right direction/explain something maybe I don't understand.

Thanks
 
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  • #2
I think multplying out as you started with is not a bad way to approach this
 
  • #3
Hmmm, maybe I did not put enough work in that direction before trying other stuff. I'll work on it and get back.

EDIT: I was wondering also, for the criteria that partial(f)/partial(zbar)=0, is zbar treated like any other varaible? In other words, since I can express the first function entirely in terms of z, can I say that partial(f)/partial(zbar) = 0 and so say that they are holomorphic?
 
Last edited:
  • #4
dh363 said:
Hmmm, maybe I did not put enough work in that direction before trying other stuff. I'll work on it and get back.

Sure, your function is e^(-z)*(z^2-2). It seems a little painful to show CR applies directly. You don't know things like the product of analytic functions is analytic??
 
  • #5
Dick said:
Sure, your function is e^(-z)*(z^2-2). It seems a little painful to show CR applies directly. You don't know things like the product of analytic functions is analytic??

mmm, i think this method and the partial(f)/partial(zbar)=0 method work to make this problem very easy. We never proved product of analytic functions is analytic, maybe he expected us to know this from analysis I. I skipped it because I wouldn't have otherwise been able to fit complex analysis into my 4 years here :( Been keeping up fine teaching myself stuff I'm not familiar with though.

lol I am an idiot, I wasn't understanding the conjugate of z equation properly. Looked back at the derivation and i get it now.
 

1. What is a complex function?

A complex function is a function that maps complex numbers to complex numbers. It can be written in the form f(z) = u(z) + iv(z), where u(z) and v(z) are real-valued functions and i is the imaginary unit.

2. What does it mean for a function to be entire?

A function is entire if it is defined and analytic (differentiable at every point) on the entire complex plane. This means that the function is smooth and has no singularities or poles.

3. How can I prove that a function is entire?

To prove that a function is entire, you need to show that it is defined and differentiable at every point on the complex plane. This can be done by using the Cauchy-Riemann equations, which state that a function is analytic if and only if it satisfies these equations.

4. What is the significance of proving a function is entire?

Proving that a function is entire is important because it guarantees that the function is well-behaved and has no singularities. This makes it easier to study and understand the behavior of the function, and allows for the use of powerful mathematical tools and techniques to solve problems involving the function.

5. Are there any common techniques for proving a function is entire?

Yes, there are several techniques that are commonly used to prove that a function is entire. These include using the Cauchy-Riemann equations, demonstrating that the function has a power series representation, and using the Cauchy integral theorem to show that the function is analytic over a closed contour.

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