# Periodic Function

A function

*f*:*X**â†’**Y*is said to be a periodic function if there exists a positive real number*p*such that*f*(*x*+*p*) =*f*(*x*), for all*x*âˆˆ*X*. The least of all such positive numbers*p*is called the principal period or simply period of*f*. All periodic functions can be analyzed over an interval of one period within the domain as the same pattern shall be repetitive over the entire domain.- sin
*x*, cos*x,*sec*x*, cosec*x*are periodic functions with period 2*Ï€*. tan*x*, cot*x*are periodic with period*Ï€*. *f*(*x*) =*x*â€“ [*x*] is periodic with period 1, where [â‹…] represents greatest integer function.

There are two types of questions asked in the examination. You may be asked to test for periodicity of the function or to find the period of the function. In the former case you just need to show that

*f*(*x*+*T*) =*f*(*x*) for same*T*(> 0) independent of*x*whereas in the latter, you are required to find a least positive number*T*independent of*x*for which*f*(*x*+*T*) =*f*(*x*) is satisfied.

Notes:

- If
*f*(*x*) is periodic with period*p*, then*a**f*(*x*+*c*) +*b*where*a*,*b, c*âˆˆ*R*(*a*â‰ 0) is also periodic with period*p*. - If
*f*(*x*) is periodic with period*p*, then*f*(*ax)*+*b*where*a*,*b*âˆˆ*R*(*a*â‰ 0) is also period with period . - Let
*f*(*x*) has period*p*=*m*/*n*(*m*,*n*âˆˆ*N*and co-prime) and*g*(*x*) has period*q*=*r*/*s*(*r*,*s*âˆˆ*N*and co-prime) and let*t*be the LCM of*p*and*q*, i.e.,*t*=*t*shall be the period of*f*+*g*provided there does not exist a positive number*k*(<*t*) for which*f*(*x*+*k*) +*g*(*x*+*k*) =*f*(*x*) +*g*(*x*), else*k*will be the period. The same rule is applicable for any other algebraic combination of*f*(*x*) and*g*(*x*).*p*and*q*always exist if*p*/*q*is a rational quantity. If*p*/*q*is irrational then algebraic combination of*f*and*g*is non-periodic. - sin
^{n}*x*, cos^{n}*x*, cosec^{n}*x,*and se*x*^{n}*x*have period 2*Ï€*if*n*is odd and*Ï€*if*n*is even. - tan
and cot^{n}x^{n}*x*have period*Ï€*whether*n*is odd or even. - A constant function is periodic but does not have a well-defined period.
- If
*g*is periodic then*fog*will always be a periodic function. Period of*fog*may or may not be the period of*g*. - If
*f*is periodic and*g*is strictly monotonic (other than linear) then*fog*is non-periodic. - Addition of periodic and non-periodic functions is always non-periodic function.
- Addition of two non-periodic functions may be periodic.