Veryfiy Picard's Theorem

  • Thread starter usn7564
  • Start date
  • Tags
    Theorem
In summary, Picard's Theorem states that a function with an essential singularity will take on every complex value, with possibly one exception, in any neighborhood of the singularity. For the function cos(1/z) at z=0, this is proven by breaking out z to get z = 1/log(c±√(c^2-1)) for an arbitrary complex constant c. This allows for values of the logarithm to be chosen to make |z|<ε for any positive ε, showing that cos(1/z) can achieve any complex number in any neighborhood of z=0. This is due to the unboundedness of the imaginary part of w=1/z and the phase degree of freedom from
  • #1
63
0

Homework Statement


Verify Picard's Theorem for cos(1/z) at z = 0

Homework Equations


The theorem:
A function with an essential singularity assumes every complex number, with possibly one exception, as a value in any neighborhood of this singularity

The Attempt at a Solution


I have a solution that I don't understand (from a manual). What the manual does is break out z to get
[tex]z = \frac{1}{log(c\pm \sqrt{c^2-1}}[/tex] for an arbitrary complex constant c. Getting there is trivial enough but I don't get the conclusion.

Values of the logarithm can be chosen to make [tex] |z| < \epsilon[/tex] for any positive [tex] \epsilon[/tex], so that cos(1/z) achieves the value c in any neighborhood of z = 0.

I understand that you can make z arbitrarily small by choosing a k high enough in the logarithm, but intuitively I'd think cos(z) wouldn't leave [-1, 1] no matter how fast it jumped between them. Intuition be damned, I can't see the reasoning at all. Because I can get a very large input in the cosine function I know the function can output any complex number? How?

Guessing it's something rather silly I'm missing here but there you go.
Any help would be appreciated, cheers.
 
Physics news on Phys.org
  • #2
I think you're missing the fact that [itex] \cos{w}[/itex] does not necessarily sit in the interval [itex] [-1,1] [/itex] when [itex] w [/itex] is complex. For example, if [itex] w [/itex] is pure imaginary, then [itex] \cos{w} = \cosh{|w|} [/itex].

So the point is that you can use the unboundedness from a large imaginary part of [itex] w =1/z[/itex] plus the phase degree of freedom from the real part of [itex] w[/itex] to get more or less any complex number you want.
 
  • Like
Likes 1 person
  • #3
You're right, that completely flew me by. Thanks, wasn't that tricky after all.
 
  • #4
usn7564 said:
How?

First, better define what you mean by Picard's Theorem. He has two, the little one and the big one.

Little: Every entire function that's not a polynomial has an essential singularity at infinity.

Big: A function with an essential singularity achieves every value, with at most one exception, infinitely often in any neighborhood of the singularity.

I think you want the big one.

Any help would be appreciated, cheers.

First do a Wikipedia on Casorati-Weierstrass, study the example for [itex]e^{1/z}[/itex], then apply that example to your problem.
 

1. What is Picard's Theorem and why is it important?

Picard's Theorem is a fundamental result in complex analysis that guarantees the existence and uniqueness of solutions to certain differential equations. It is important because it provides a powerful tool for solving many mathematical problems in fields such as physics, engineering, and economics.

2. Who is Picard and when was his theorem discovered?

Jules Henri Poincaré, also known as Henri Poincaré, is the mathematician who first proved Picard's Theorem in 1890.

3. What types of equations can be solved using Picard's Theorem?

Picard's Theorem can be used to solve ordinary differential equations with initial conditions. It can also be extended to partial differential equations with initial conditions and certain boundary conditions.

4. Are there any limitations to Picard's Theorem?

Yes, Picard's Theorem only applies to equations with continuous and locally Lipschitz continuous coefficients. This means that some equations with discontinuous coefficients may not have a solution guaranteed by the theorem.

5. How is Picard's Theorem used in real-world applications?

Picard's Theorem has many practical applications in fields such as engineering, physics, and economics. It is used to solve differential equations that model physical phenomena, such as the motion of particles in a fluid or the growth of a population over time. It is also used in numerical methods for approximating solutions to differential equations.

Suggested for: Veryfiy Picard's Theorem

Replies
3
Views
342
Replies
3
Views
527
Replies
26
Views
1K
Replies
23
Views
1K
Replies
7
Views
807
Replies
2
Views
549
Replies
8
Views
786
Replies
1
Views
532
Replies
6
Views
798
Back
Top