True or False? (Complex Analysis)

In summary, the conversation discusses the properties of holomorphic functions on a star-shaped open subset of the complex plane, where f is a holomorphic function from S to \mathbb{C} and z_0 is an element of S. The first statement states that if g is a holomorphic function on S except for a pole of order N at z_0, then the residue of g at z_0 is given by the coefficient a_{-1} in its Laurent Series. The second statement states that if S is a disk centered at z_0 and f is a holomorphic function on S, then the coefficients of its Taylor series at z_0 are equal to f(z_0) and its derivatives at z_
  • #1
Ted123
446
0
[itex]S[/itex] is a star-shaped open subset of [itex]\mathbb{C}[/itex], [itex]f[/itex] is a holomorphic function from [itex]S[/itex] to [itex]\mathbb{C}[/itex], [itex]z_0[/itex] is an element of [itex]S[/itex].

I've just come out an exam and wondered whether the following 2 statements are true or false:

1 Let [itex]g[/itex] be a holomorphic function on [itex]S \subseteq \mathbb{C}[/itex], with the exception of a pole of order [itex]N[/itex] at [itex]z_0[/itex]. If the Laurent Series of [itex]g[/itex] around [itex]z_0[/itex] is

[itex]\displaystyle \sum_{n=-N}^{\infty} a_n ( z - z_0 )^n[/itex]

for [itex]z \in D'(z_0, R)[/itex] for some [itex]R>0[/itex] (and [itex]D(z_0 , R) \subseteq S[/itex]) and constants [itex]a_n \in \mathbb{C}[/itex], then the residue of [itex]g[/itex] at [itex]z_0[/itex] is given by [itex]a_{-1}[/itex].

2 Suppose [itex]S = D(z_0, R)[/itex] for some [itex]R>0[/itex] and

[itex]\displaystyle f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n[/itex]

for all [itex]z \in S[/itex] and some constants [itex]a_n \in \mathbb{C}[/itex]. Then necessarily [itex]a_0 = f(z_0)[/itex] and [itex]a_n = f^{(n)}(z_0)[/itex] for all [itex]n\geq 1[/itex].
 
Physics news on Phys.org
  • #2
1. How are the coefficients of the Laurent Series defined? (hint: it is related to the Cauchy Integral Formula)

2. Your function is holomorphic on S. So it is equal to its Taylor series on any point in its domain. What should the coefficients of a Taylor series centered at z_0 be?
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex variables. It involves the application of calculus and algebra to complex numbers, which are numbers that have both real and imaginary components.

2. What is the difference between true and false in complex analysis?

In complex analysis, true and false refer to statements that can either be proven to be true or false using mathematical methods. True statements are those that are logically and mathematically valid, while false statements are those that are not supported by evidence or contradict known facts.

3. How are true and false statements used in complex analysis?

True and false statements are used to prove theorems and make deductions in complex analysis. By determining the truth value of a statement, mathematicians can build upon existing knowledge and discover new results in the field.

4. Is there a gray area between true and false in complex analysis?

In complex analysis, there is a concept known as "undecidability," where a statement cannot be proven to be either true or false using current mathematical methods. This gray area highlights the limitations of human knowledge and the need for further research and development in the field.

5. What are some real-world applications of true and false in complex analysis?

Complex analysis has many practical applications, including in physics, engineering, and economics. For example, it is used in the study of fluid dynamics, electrical circuits, and stock market predictions. By understanding the truth value of complex statements, scientists and engineers can make accurate predictions and create more efficient systems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
812
  • Topology and Analysis
Replies
14
Views
340
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
532
  • Calculus and Beyond Homework Help
Replies
2
Views
152
  • Calculus and Beyond Homework Help
Replies
1
Views
450
  • Calculus and Beyond Homework Help
Replies
3
Views
466
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Math POTW for Graduate Students
Replies
2
Views
758
  • Calculus and Beyond Homework Help
Replies
14
Views
429
Back
Top