Odd/even functions and periodicity

  • Thread starter Thread starter dimitri151
  • Start date Start date
  • Tags Tags
    Functions
Click For Summary
SUMMARY

The discussion centers on the relationship between odd and even functions, specifically addressing the theorem that if f(x) is an odd function and f(x + t) is an even function, then f(x) is periodic with a period of at most 4t. The example provided is f(x) = sin(x), where it is demonstrated that with t = π/2 and n = 4, the function exhibits periodicity. The conclusion drawn is that if f(x) is odd and f(x + t) is even, then f(n2t) = 0 for all integers n, confirming the periodic nature of the function.

PREREQUISITES
  • Understanding of odd and even functions
  • Familiarity with periodic functions and their properties
  • Knowledge of trigonometric functions, specifically sine
  • Basic grasp of mathematical proofs and definitions
NEXT STEPS
  • Research the properties of odd and even functions in more depth
  • Explore the concept of periodicity in trigonometric functions
  • Study the implications of Fourier series on function periodicity
  • Investigate other theorems related to odd/even functions and their periodicity
USEFUL FOR

Mathematicians, students studying calculus or advanced mathematics, and anyone interested in the properties of functions and their periodic behavior.

dimitri151
Messages
117
Reaction score
3
You can prove if f(x) is an odd function and f(x+ t) is an even function then f(x) is periodic with period at most 4t. Are there other theorems like that?i know this is a somewhat open ended and general question, it's just i would like to squeeze some more results from this angle and can not.
 
Physics news on Phys.org
i would like to squeeze some more results from this angle and can not.
... what "more results"? "more" suggests you have already got some results. What results have you got so far?

Consider the specific example where f(x)=sin(x).
The general approach would start with the definitions of both.

(modify slightly so that f(x-t) is even, makes it easier to write...)

if f(-x)=-f(x) and f(t-x)=f(x-t) then f(x)=f(x-nt): n in Z (?)

for f(x)=sin(x), t=pi/2, n=4.
 
The result so far is that if f(x) is odd and f(x+t) is even then f(n2t) =0 for all integer n, f(x) is periodic, and the minimum period is no greater than 4t.
 

Thread 'How does this derivative work? And why the speed of light remains?'

Relativistic Momentum, Mass, and Energy Momentum and mass (...), the classic equations for conserving momentum and energy are not adequate for the analysis of high-speed collisions. (...) The momentum of a particle moving with velocity ##v## is given by $$p=\cfrac{mv}{\sqrt{1-(v^2/c^2)}}\qquad{R-10}$$ ENERGY In relativistic mechanics, as in classic mechanics, the net force on a particle is equal to the time rate of change of the momentum of the particle. Considering one-dimensional...
View full post »

Similar threads

Undergrad Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines
  • · Replies 139 ·
5
Replies
139
Views
11K
Undergrad Uniform convergence and derivatives -- difference between two theorems?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Multivariable fundamental calculus theorem in Wald
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Open interval or Closed interval in defining convex function
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Doubt about theorem in Calculus on Manifolds
  • · Replies 9 ·
Replies
9
Views
3K
High School Functions which relate to calculus: Questions about Notation
  • · Replies 36 ·
2
Replies
36
Views
6K
High School Question about the fundamental theorem of calculus
  • · Replies 2 ·
Replies
2
Views
3K