A \(sequence\) is a function \(f\) defined for every non-negative integer n. For sequences, generally it is set that \(x_n=f(n)\). Usually we are given equation of the form
Sometimes we are expected to find a "closed expression" for . Such an equation is called . A function equation of the form
is a (homogeneous) (with constant coefficients).
To find the general solution of , first we try to find a solution of the form for a suitable number . To find , we plug into and get
And is called the of . For distinct roots ,
is the general solution. can be found from the initial values and .
If , the general solution has the form
Now we will talk about the functional equation of function (non-zero) of the form
No it looks like a linear difference equation of order 2. But the discrete variable is replaced by the continuous variable . So we try to find the solutions . For the value of , we get
with the solutions
For , we have the solutions
So and its conjugate are unit vectors in the complex plane, that is ,
Thus has a period , if
Putting the value above, we get
Therefore Fundamental period for non-zero functions satisfying .
Yet it is unlikely that this irrational number gives a rational multiple of for the angle , the only way to secure periodicity.
You can try the problems based on it problem 1, and problem 2
You can try more such problems of the set Do you know its property ?