Recursive functions of functions in the field of mathematics

We know that there will be recursive calls to a function in the field of programming, and we can sometimes turn to non-recursive calls for recursive calls, and the purpose of converting to non-recursive calls can reduce the allocation of stack memory and reduce the memory overflow problem. So in the general field of mathematics we also have some recursive invocation of the situation, such as in the analog circuit of the feedback circuit, the output of a calculation once again as the input of the calculation. Therefore, the study of recursive function has practical significance.

Suppose there is a function y =? (x) where x is the same as the domain of the Y. Suppose there is y0= when x=x0? (x0), when we put y0 the most time as the argument and substituting? (x) to go to the y1=? (y0), then Y1 can also be expressed as:

Y1 =? (? (x0))

We call Y1 the value that x0 has received after 2 function calls. At the same time we call g (x) =? (? (x)) is a function? (x) two recursive functions. We can define the following:

Definition? n (x) is? (x) a function obtained after n recursion, referred to as n-time recursive function, where n>=1, and we stipulate? 1 (x) =? (x)

Then there are several problems to be faced, whether any one function has n recursive function, if there are n recursive function, then can a non-recursive function to describe the recursive function? and is the value obtained after the M-recursion of a function equal to the first argument? And what is the inverse function of a function related to a recursive function?

Some of the above questions want to discuss and study with like-minded classmates.

For some elementary functions, we can easily get the N-times recursive function of a function, for example:

y = a*x +b's n recursive function is y = a^n *x + b*∑a^i (where i=0 to n-1)

y = n recursive function for x^a = y = x^ (a^n)

y = 1/x's n-th recursive function is y = ((x^2-1) * ( -1) ^n + x^2 + 1)/2*x (this formula is not as simple as that, the subsequent derivation process is given)

Some functions, such as

y = sin (x) We have no way to represent its n-times recursive function with a function that can be defined (is it really not?). Still needs to be studied), it can only be represented by the enumeration method for the time being:

y = sin (sin (sin (...). (x) ...))

Recursive cycle of functions

Suppose a function? (x) N-time recursive function? n (x) when N=M1,M2,MK? MI (x) = x WHERE (i = 1 to k) then we call the minimum MI as the recursive period of the function, called M.

The next question is how to determine whether an n-recursive function has a recursive period and how the recurrence cycle is calculated, and this is the subject of research.

2-Time Recursive periodic function

If a function has a recursive period of 2 it is called the function as a recursive periodic function of 2 times. The 2-time recursive periodic function is actually y=f (x) when n is the cardinality, and when n is an even number is y=x, that is, the recurrence period is as follows:

F1 (x) = f (x), f2 (x) = X, F3 (x) =f (x), f4 (x) = x, f5 (x) = f (x),.....

How to deduce an expression of a 2-time recursive periodic function.

We examine a function y = ( -1) ^ n where n∈n. The result of this function is that when n is even, the result is 1, and when n is odd, the result is-1. So we can define a linear function y = ax+b. and assume that when X=-1 y=a, when X=1 y=b. Then the analytic formula for this linear function is:

y = (b-a)/2 * x + (B+A)/2 where a= (b-a)/2, B = (b+a)/2

Since x can only be 1, and 1 above can be true, we redefine the function as follows:

y = (b-a)/2 * ( -1) ^n + (B+a)/2 where n∈n, the image of this function is when n is odd when y=a, and when n is even, y = B.

We then analyze from the above function 2 times recursive periodic function to find that when n is odd, fn (x) = f (x) and when n is even, fn (x) = x So we can conclude that the function of 2 recursive periodic function is:

fn (x) = (x-f (x))/2 * ( -1) ^n + (x+ f (x))/2

Full recursive function

We know that if a function is Y =f (x), then the inverse function of the function is represented by x = F-1 (y). If you recursively return this inverse function

。。。

Because we define as follows:

If a function y=f (x) n-times the recursive function of FN (x) and when n = 0 is F0 (x) = x, and the N-times inverse recursive function is GN (x) Then we call the function

fn (x) n>=1

KN (x) = {x n = 0

GN (x) n < 0

is the full recursive function of f (x).

And if the n-times recursive function of a function is fn (x) when n = 0 o'clock has F0 (x) = x; And there are N < 0 o'clock fn (x) = GN (x) then we call FN (x) the only function of the full recursive function, and call this full recursive function a unique full recursive function.

Note that not all functions of the n-th recursive function are all recursive functions.

y = a * n recursive function of x is y= a^n*x, and the full recursive function is y = a^n*x

y = (x+1)/(X-1) n-times recursive function is y = (( -1) ^ (n+1) + 2x + 1)/((2-x) * ( -1) ^n + x) full recursive function is y = (( -1) ^ (n+1) + 2x + 1)/((2-x) * ( -1) ^n + x)

y = 1/x N-Times recursive function is y = ((x^2-1) * ( -1) ^n + x^2 + 1)/2*x full recursive function is y = ((x^2-1) * ( -1) ^n + x^2 + 1)/2*x

y = x^a N-Times recursive function is y = x^ (a^n) Full recursive function is y = x^ (a^n)

y = a*x + b n times the recursive function is y = a^n*x + b *∑a^ (i-1) (i = 1 to n) The full recursive function is:

a^n*x + b *∑a^ (i-1) (i = 1 to n) N>=1

y = {x n = 0

A^n*x-b * ∑a^i (i =-1 to-n) n < 0

Recursive functions of functions in the field of mathematics