Periodic Functions: Is There a Non-Trig Function?

In summary, there are many continuous periodic functions that are not trigonometric. One example is f(x) = Arccos(cos(x)) = x, which is periodic with period pi. Other examples include using the principle value of Arccos to write a general periodic function, and using modulus or raising any real number to the power of any other real number plus an imaginary variable. Additionally, all functions are technically "man-made," but not all are explicitly constructed by humans. Some examples of continuous periodic functions that are not explicitly constructed include constant functions and the function f(x) = x-2n for x belonging to [2n, 2n+1) where n is any integer.
  • #1
geniusprahar_21
28
0
Is there a continuous periodic function which is not trigonometric. if yes, what?
 
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  • #2
geniusprahar_21 said:
Is there a continuous periodic function which is not trigonometric. if yes, what?
Yes there are very many. Define a continuous function on [a,b] where f(a)=f(b) then define f outside of [a,b] so that f(x+(b-a))=f(x). A simple example that is not trigonometric (even though it looks like it is) is Arccos(cos(x)).
 
  • #3
f(x) = Arccos(cos(x)) = x, the identity function is periodic. Now besides this one and the trig functions, are there other non "man-made" (i.e. cut and pasted according to the process described by lurflurf) that are periodic?
 
  • #4
I don't know any functions that aren't "man-made"!
 
  • #5
...f(x) = Arccos(cos(x)) = x...

No, the inverse cosine function returns values in a specific interval (which I can't remember atm), so you can't have arccos(cos(x)) = x for all x.
 
  • #6
Muzza said:
No, the inverse cosine function returns values in a specific interval (which I can't remember atm), so you can't have arccos(cos(x)) = x for all x.
That is right Arccos(cos(x))=x on [0,pi], it is also periodic with period pi, so it repeats all those values. I use Arccos with the capital A to make clear that I am using the principle value of Arccos not just any value that gives the needed value. This is a general way to write periodic functions. let f(x) be diffined and continuous on [a,b] with f(a)=f(b) then
g(x)=f(a+(b-a)(1+(1/pi)Arccos(cos(pi(x-a)/(b-a)))))
is a periodic extension of f that is f=g on [a,b] and g(x+2n(b-a))=g(x)
when n is an integer.
remenber the definition of a periodic function is a function is periodic with period p if
f(x+p)=f(x) for all x.
 
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  • #7
Modulus is periodic, any real number to the power of any other real number + an imaginary variable is periodic.

For example, there is:
e^(2+x*i)
 
  • #8
HallsofIvy said:
I don't know any functions that aren't "man-made"!
Every function is man-made. Not every function is "man-made". :wink:
 
  • #9
every function is man-made yaar...mathematics itself is man-made ;) functions are infinite...i can define a function rite now 2 suit ur needs...lemme see...
f(x)=x-2n for x belonging to [2n, 2n+1) where n is any integer
= (2n+2)-x for x belonging to [2n+1, 2n+2]
check this out...if i haven't made any silly mistakes...it shud come out 2 be continuous and periodic...ive modeled it on the sin graph + on the [x] graph..lol...cudnt think of a better example sorry...cheers! ;)
 
  • #10
Constant functions.
 

1. What is a periodic function?

A periodic function is a function that repeats its values at regular intervals, called the period. This means that the same output values occur at regular intervals as the input values change.

2. Can a function be periodic without involving trigonometric functions?

Yes, a function can be periodic without involving trigonometric functions. A simple example is the sawtooth wave function, which repeats its values in a linear pattern. Other examples include square waves, triangle waves, and step functions.

3. How can we determine if a function is periodic?

To determine if a function is periodic, we need to find a constant value, called the period, which when added to or subtracted from the input value, gives the same output value. If such a constant exists, then the function is periodic. Otherwise, the function is not periodic.

4. Is there a non-trigonometric function that can model periodic phenomena?

Yes, there are many non-trigonometric functions that can model periodic phenomena. For example, the logistic function can be used to model population growth, and the sine logistic function can be used to model the spread of infectious diseases. These functions exhibit periodic behavior without involving trigonometric functions.

5. Are there any benefits to using non-trigonometric functions to model periodic phenomena?

Yes, there are benefits to using non-trigonometric functions to model periodic phenomena. Non-trigonometric functions can provide a more accurate representation of real-world phenomena, as they can capture more complex and irregular periodic patterns. Additionally, non-trigonometric functions may be easier to work with and analyze mathematically compared to trigonometric functions.

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