Question

I need to know the name of a function that's 1 at every integer and 0 everywhere else. Anyone have any info on a function like this?

Answer 1

HI!!

Answer 2

they call this kind of thing a "characteristic function"

Answer 3

this would be the characteristic function of the integers

Answer 4

Should be nice to see, I'll look into that. Here's some alternative definition I came up with among others to help me out in the mean time: \[\Large f(x) = \lim_{r \to \infty} \cos^r(\pi x)\]

Answer 5

@misty1212 Hey could you give me some more information, I'm sorta having trouble in google searching "Characteristc Function" haha

Answer 6

should be possible with floor/ceil : \[f(x) = -\lfloor x\rfloor -\lfloor -x\rfloor \]

Answer 7

That's something I hadn't thought of that looks pretty cool! The reason I'm interested in this function is because: \[\Large \tau (x) = \sum_{n=1}^{ \infty } f(\frac{x}{n})\]

Answer 8

that gives 0 when x is integer and 1 otherwise

Answer 9

(refering to floor function)

Answer 10

Oh so I need 1-that I think? I'm going to play with it and just figure it out one sec.

Answer 11

Here's what I believe/am saying right now. \[\Large \lim_{r \to \infty} \cos^r( \pi x) = 1+ \lfloor x \rfloor + \lfloor -x \rfloor\]

Answer 12

I typically call them indicator functions

http://en.wikipedia.org/wiki/Indicator_function

when I hear characteristic function I think of this

http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29

Answer 13

Ahhh good and it's related to the Heaviside and Delta functions, thanks this is something I think I can work with now.

Answer 14

\[\mathbb{1}_{\mathbb{Z}}(x)\]

Answer 15

Interesting, I think this might be the first time I've seen a function denoted with a number instead of a letter before haha.

Answer 16

what is wrong with plain old \[f(x) = \left\{\begin{array}{rcc}
1 & \text{if} ~x\in\mathbb{Z} \\
0& \text{otherwise}
\end{array}
\right.
\]

Answer 17

that would require one to not be lazy

Answer 18

I'm too lazy to write that much

Answer 19

how about "one if x is an integer, zero otherwise"?

Answer 20

That seems like a lot of words to me. I don't like to write that much when I'm on vacation.