# Name for sine-like functions

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Homework Helper
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?

Stephen Tashi and Delta2

anuttarasammyak
Gold Member
As an example
$$f(x)=e^{ax}$$as
$$f^{(n+1)}(x)=a f^{(n)}(x)$$so
$$\alpha_n=a$$ otherwise $$\alpha_k=0$$ There are other expressions, e.g.,
$$\alpha_k=\frac{1}{n+1}a^{n-k+1}$$
The component functions are all the same.

Homework Helper
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.

Office_Shredder
Staff Emeritus
Gold Member
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.

$$\sin(x)=\frac{ e^{ix} - e^{-ix}}{2}.$$

Stephen Tashi