Name for sine-like functions

  • B
  • Thread starter Gaussian97
  • Start date
  • #1
Gaussian97
Homework Helper
593
334
Summary:
Is there a name for functions that are linearly dependent with its derivatives?
Is there a name for functions that are linearly dependent with its derivatives? i.e. a function ##f(x)## such that, for some value of ##n## it fulfills
$$f^{(n+1)} = \sum_{k=0}^{n} \alpha_k f^{(k)}$$
are linearly dependent?
 
  • Like
Likes Stephen Tashi and Delta2

Answers and Replies

  • #2
anuttarasammyak
Gold Member
819
354
As an example
[tex]f(x)=e^{ax}[/tex]as
[tex]f^{(n+1)}(x)=a f^{(n)}(x)[/tex]so
[tex]\alpha_n=a[/tex] otherwise [tex]\alpha_k=0[/tex] There are other expressions, e.g.,
[tex]\alpha_k=\frac{1}{n+1}a^{n-k+1}[/tex]
The component functions are all the same.
 
  • #3
Gaussian97
Homework Helper
593
334
Yes, if we define the set ##D_k## as the set of all such functions with ##n=k-1##, then any exponential would be on ##D_1## because the first derivative is LD to the function itself, ##\sin{x}## and ##\cos{x}## would be on ##D_2## because the second derivative is LD to the function, and a polynomial with degree ##j## would be on ##D_{j+1}##. But I was asking if there is a special name given to such functions.
 
  • #4
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
4,551
586
I don't think you get polynomials in this set, the top degree gets destroyed by derivatives so you can never get rid of it with a linear equation in the derivatives.

I don't know of a name for these. I suspect they are all secretly linear combinations of the exponential function with complex arguments, e.g.

$$\sin(x)=\frac{ e^{ix} - e^{-ix}}{2}.$$
 
  • #5
Stephen Tashi
Science Advisor
7,664
1,500
Summary:: Is there a name for functions that are linearly dependent with its derivatives?
Is this equivalent to asking about all functions that are solutions to some homogeneous linear differential equation with constant coefficients?
 

Related Threads on Name for sine-like functions

Replies
1
Views
587
Replies
6
Views
21K
Replies
9
Views
6K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
3K
Replies
3
Views
2K
Replies
2
Views
2K
Replies
2
Views
1K
Replies
1
Views
3K
Top