Uniqueness of Analytic Functions on a Disc

In summary, the mathematician sought a unique analytic function satisfying the condition f(z) is analytic in the disc |z-1| < 1 and also has a zero at z=1 which is isolated. He found that the function is f(z) = 2z / (1+z^2).
  • #1
nicksauce
Science Advisor
Homework Helper
1,271
7

Homework Statement


Find all functions f(z) satisfying a) f(z) is analytic in the disc |z-1| < 1, and b) f(n/(n+1)) = 1 - 1 / (2n^2 + 2n+1).

Homework Equations


The Attempt at a Solution


One can deduce by algebraic re-arrangement that one solution is f(z) = 2z / (1+z^2). But how can I show that this function is the unique function satisfying this condition? Any hints?
 
Physics news on Phys.org
  • #2
Let g(z)=2z/(1+z^2). Then let h(z)=f(z)-g(z). The zeros of h(z) have a cluster point at z=1. The zeros of an analytic function which is not zero are isolated. So what is h(z)?
 
  • #3
Hi Dick, thanks for the reply. A few things: If you let g(z) = 2z/(1+z^2) and you have already f(z) = 2z/(1+z^2), then isn't h(z) identically 0? I don't see what you meant here.

Second, this was from a fairly unrigorous class intended for physicists, so we never defined 'cluster point'. Perhaps you could re-phrase in more intuitive language?
 
  • #4
Hi nicksauce,

You don't have f(z)=2z/(1+z^2), that's what you are trying to prove, isn't it? And proving h(z)=0 will do that. Sorry to use words without defining them, it didn't look like a physics question. g(1)=1 and by continuity f(1)=1. So h(1)=0. If h(z) is analytic and nonzero, and it IS analytic since f(z) and g(z) are, then the zero at z=1 must be isolated. Which means there is a disk around z=1 where h(z) is nonzero. Since the limit of n/(n+1) is 1 (i.e. 1 is a cluster point) and h(n/(n+1))=0, that contradicts h(z) having an isolated zero at z=1. Hence, h(z)=0.
 
Last edited:
  • #5
Ok that seems to make sense. Thanks!
 
  • #6
nicksauce said:
Ok that seems to make sense. Thanks!

'seems to make sense'?? REALLY!?? It does make sense! :). I don't know exactly what field of physics you're in, but there is a surprising amount of info you can get out of complex analysis. Look at the S-matrix. If you assume it's analytic with branch cuts in the complex energy plane you get all kinds of stuff.
 
  • #7
Allow me to clarify a few things...

Dick said:
it didn't look like a physics question.
I didn't mean it was physics questions. What I meant was that as a physics student, I took the complex variables course that is intended for physics students, thus with more of a focus on calculations and applications and a bit of a lack of rigorous definitions, rather than a course intended for math students.

Dick said:
Look at the S-matrix. If you assume it's analytic with branch cuts in the complex energy plane you get all kinds of stuff.
Well I am just an undergrad and I have no idea what an S-matrix is, but I'll try to keep that in mind. :p
 

1. What is the importance of uniqueness of function in scientific research?

The uniqueness of function refers to the specific role or purpose of a biological or chemical component within a system. It is important in scientific research because understanding the unique function of each component allows for a better understanding of the overall system and its interactions. This information can then be used to make accurate predictions and develop effective solutions in various fields such as medicine, agriculture, and environmental science.

2. How do scientists determine the uniqueness of function in a biological system?

Scientists use a variety of methods to determine the uniqueness of function in a biological system. This can include experiments, genetic studies, and computer simulations. By manipulating and observing different components within the system, scientists can identify their specific functions and how they contribute to the overall function of the system.

3. Can the uniqueness of function change over time?

Yes, the uniqueness of function can change over time. Biological systems are constantly evolving and adapting to their environment, and as a result, the functions of different components within the system can also change. This is especially true for organisms that undergo genetic mutations or environmental changes.

4. How does understanding the uniqueness of function contribute to the development of new technologies?

Understanding the uniqueness of function is crucial in the development of new technologies. By studying the specific functions of different components within a system, scientists can identify which components are essential for a particular function and how they can be manipulated or replicated for practical use. This knowledge has led to advancements in various fields such as medicine, biotechnology, and renewable energy.

5. What are some challenges in determining the uniqueness of function in complex systems?

One of the main challenges in determining the uniqueness of function in complex systems is the interconnectedness of different components. In many cases, one component may have multiple functions and can also interact with other components in various ways. This makes it difficult to isolate and identify the specific function of each component. Additionally, some components may have unknown or undiscovered functions, making it challenging to determine their uniqueness. Sophisticated techniques and advanced technologies are continually being developed to overcome these challenges.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
356
  • Calculus and Beyond Homework Help
Replies
2
Views
221
  • Calculus and Beyond Homework Help
Replies
6
Views
695
  • Calculus and Beyond Homework Help
Replies
6
Views
274
  • Calculus and Beyond Homework Help
Replies
8
Views
347
  • Calculus and Beyond Homework Help
Replies
7
Views
510
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
503
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top