Functions that vanish at integers

In summary, the conversation discusses the existence of functions that vanish at all integers, and the possibility of generalizing this concept to higher dimensions. The conversation also touches on the idea of a function being defined as a formula and the potential complexity of such functions.
  • #1
imAwinner
10
0
We know that the function f(x) = sin(2*pi*x) vanishes at all integers, are there other functions like that and what is the appropriate generalization to higher dimensions?

Cheers
 
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  • #2
This is probably not what you are looking for, but unless you define a function like; f is a function that is equal to some constant y for non-integer values, and 0 for integer values; then you will find there is a very limited range of functions you can have.

You could also just put an integer K into the argument on the sine. And put any constant outside the sine.
 
  • #3
There are as many as you can draw. And more.
 
  • #4
ZioX said:
There are as many as you can draw. And more.

i don't no... i can draw like a hundred of those... and id ont think there are more than that...









:tongue:
 
  • #5
Maybe I wasn't specific, I'm looking for the formula of a function f(x,y) that would vanish at integers, could anyone help?
 
  • #6
imAwinner said:
We know that the function f(x) = sin(2*pi*x) vanishes at all integers, are there other functions like that and what is the appropriate generalization to higher dimensions?

For one-dimensional functions [itex]f(x): \mathbb{R}\to\mathbb{R}[/itex], there are
[tex]\beth_1^{\beth_1}=\beth_2=2^{2^{\aleph_0}}[/tex] ("lots")
of functions that vanish on the integers.
 
  • #7
imAwinner said:
Maybe I wasn't specific, I'm looking for the formula of a function f(x,y) that would vanish at integers, could anyone help?

Suppose we had two such real valued functions on the reals. What new function could you create such that its restriction on Z^2 is identically zero and nonzero on R^2/Z^2?
 
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  • #8
First of all note that you haven't required that the functions be continuous so you can pretty much pick any values you want for non-integer x and require f(n) be 0 for integer n. How about f(x)= 1 if x is not an integer, 0 if x is an integer?

Even requiring continuous, it is always possible to be whatever (constant) value you want between n and -n and and add a section for x between [itex]n- \epsilon[/itex] and n (for very small [itex]\epsilon[/itex] to drop continuously to 0 at n. And there is certainly no reason to require that f not be very complicated between integers.

If you don't think there are more than "hundreds" of such functions (or if you think of functions as limited to "formulas") then you have a very restricted idea of "function".
 
  • #9
HallsofIvy said:
If you don't think there are more than "hundreds" of such functions (or if you think of functions as limited to "formulas") then you have a very restricted idea of "function".

Before there's any confusion, the "hundred" comment is mine (in jest).
 

1. What are functions that vanish at integers?

Functions that vanish at integers are mathematical functions that have a value of zero when the input is an integer. In other words, the function has no output or "vanishes" at whole numbers.

2. What is an example of a function that vanishes at integers?

One example is the sine function, sin(x), which has a value of zero at all integer values of x, such as 0, 1, -2, etc.

3. Why is it important to study functions that vanish at integers?

These functions have unique properties and can be used to solve various problems in mathematics, physics, and engineering. They also provide insights into the behavior of more complex functions.

4. How can functions that vanish at integers be graphed?

These functions can be graphed by plotting points where the input is an integer and the output is zero, and then connecting these points with a smooth curve. The resulting graph will have a series of "dips" or "peaks" at each integer value.

5. Can functions that vanish at integers have other values besides zero at non-integer inputs?

Yes, these functions can have non-zero values at non-integer inputs. However, their values at integer inputs will always be zero.

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