Eigenfunctions and eigenvalues

In summary, the conversation discusses how to solve the problem of finding the solution for f in the equation \frac {d^2} {d \phi^2} f(\phi) = q f(\phi). One approach is to "guess" a solution using the standard method of replacing f with e^{r\phi} and solving for the unknown, r. The solution is then plugged back into the original equation. The conversation also mentions another approach of replacing f'' with r^2, f' with r, and f with 1, and solving for r. The result is the same in both approaches. The conversation also touches on the topic of periodic functions and how the eigenvalues of the equation are related to the solution
  • #1
cyberdeathreaper
46
0
This is probably a straight forward question, but can someone show me how to solve this problem:

[tex]
\frac {d^2} {d \phi^2} f(\phi) = q f(\phi)
[/tex]

I need to solve for f, and the solution indicates the answer is:
[tex]
f_{\substack{+\\-}} (\phi) = A e^{\substack{+\\-} \sqrt{q} \phi}
[/tex]

I know I've covered this before - just need a refresher on how to solve.
 
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  • #2
One way to solve this problem is to "guess" a solution. Since the form of your equation is very basic, this guess is usually taught as a standard method.

As your solution suggests, your guess should be the following:
[tex]f(\phi)=e^{r\phi}[/tex]
Simply replace every function f with that equation above and solve for the unknown, r. Then, once you find r, plug it back into the equation above. Note that the "A" in your answer is an arbitrary constant used to solve with initial conditions.
 
  • #3
thanks - knew it was something simple. I actually remembered the other approach too, where you replace f'' with r^2, f' with r, and f with 1, and then solve for what r is. But either approach gives the same result.

Thanks again though.
 
  • #4
An additional question, somewhat related:

When determining the eigenvalues, the problem indicates that
[tex]
f (\phi + 2\pi) = f (\phi)
[/tex]

Given the answer already shown, why would this periodic function require:
[tex]
2 \pi \sqrt{q} = 2 n \pi i
[/tex]
 
  • #5
Take your given condition and substitute in the solution that has been found. Write the exponent of a sum as a product of exponents and the desired result follows automatically.

Do it. If you have trouble, show us what you tried and what part is bothering you.
 
  • #6
Nevermind, I got it now - didn't realize the relation between 1 and e^(i2n(pi))...

[tex]
A e^{\sqrt{q} \phi} = A e^{\sqrt{q} \left( \phi + 2 \pi \right)}
[/tex]
[tex]
e^{\sqrt{q} \phi} = e^{\sqrt{q} \phi} e^{\sqrt{q} 2 \pi}
[/tex]
[tex]
1 = e^{\sqrt{q} 2 \pi}
[/tex]
[tex]
e^{i 2 n \pi} = e^{\sqrt{q} 2 \pi}
[/tex]
[tex]
i 2 n \pi = \sqrt{q} 2 \pi
[/tex]
[tex]
i n = \sqrt{q}
[/tex]
[tex]
q = -n^2 (n=0,1,2...)
[/tex]
 
Last edited:

1. What is an eigenfunction?

An eigenfunction is a special type of function that remains unchanged when multiplied by a scalar value. It is a fundamental concept in linear algebra and is commonly used in the study of differential equations.

2. What is an eigenvalue?

An eigenvalue is a scalar value that is associated with an eigenfunction. It represents the amount by which the eigenfunction is scaled when multiplied by that value.

3. How are eigenfunctions and eigenvalues related?

Eigenfunctions and eigenvalues are closely related, as an eigenvalue is always associated with an eigenfunction. The eigenvalue determines the scaling factor of the eigenfunction, while the eigenfunction helps to solve for the eigenvalue.

4. What is the significance of eigenfunctions and eigenvalues?

Eigenfunctions and eigenvalues are important in mathematics and physics because they provide a way to simplify complex problems. They are used to solve differential equations, analyze the behavior of physical systems, and understand the properties of matrices.

5. How are eigenfunctions and eigenvalues used in real-world applications?

Eigenfunctions and eigenvalues have a wide range of applications in various fields such as signal processing, quantum mechanics, and computer graphics. They are also used in data analysis and pattern recognition to identify important features and reduce the dimensionality of data.

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