Understanding Odd, Periodic Functions: Integrals and Periodic Shifts

In summary, the conversation discusses two statements related to odd, periodic functions of period 2L. The first statement states that integrating such a function over one period will result in a zero integral. The second statement states that adding a constant to a function with period 2L will not change its period. After some initial confusion, it is determined that the second statement is obvious and the first statement can be proven by using the previous result.
  • #1
Niles
1,866
0

Homework Statement


Hi all.

Can you confirm these statements:

1) If I integrate an odd, periodic function of period 2L over one period, then the integral equals zero.

2) If I have a function f(x) with period 2L, then f(x+alfa), where alfa is an arbitrary number, will not change it's period.

Thanks in advance.


Niles.
 
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  • #2
Niles said:
Can you confirm these statements:

1) If I integrate an odd, periodic function of period 2L over one period, then the integral equals zero.

2) If I have a function f(x) with period 2L, then f(x+alfa), where alfa is an arbitrary number, will not change it's period.

Hi Niles! :smile:

(have an alpha: α :smile:)

Hint: prove 2) first (use the obvious substitution) … then use that result to prove 1). :wink:
 
  • #3
Thanks for the α.

Hmm, well, does #2 really need any proof? I mean, isn't it kinda obvious? Adding a constant α will just translate the function on x-axis (either to the left or right), so the period will remain unchanged.

And for #1: I can't see the link between this and the previous question.Niles.
 
  • #4
oops!

Hi Niles! :smile:
Niles said:
Hmm, well, does #2 really need any proof? I mean, isn't it kinda obvious? Adding a constant α will just translate the function on x-axis (either to the left or right), so the period will remain unchanged.

And for #1: I can't see the link between this and the previous question.

oops!

I misread 2) as ending "will not change its integral" :redface:

Prove that, and then prove 1).

Sorry! :smile:
 
  • #5
I have proven that the definite integral of a 2L-periodic function is the same over any interval of length 2L.

But I still can't see what the link is between this proof/theorem and my question #1.
 
  • #6
Ok, I got it now.. you were right.

Thanks!
 

FAQ: Understanding Odd, Periodic Functions: Integrals and Periodic Shifts

What are odd, periodic functions?

Odd, periodic functions are mathematical functions that have the property of being symmetric about the origin (x=0) and repeating themselves at regular intervals in both positive and negative directions. They can be represented by graphs that are symmetrical about the origin and have a repeating pattern.

How can odd, periodic functions be identified?

Odd, periodic functions can be identified by their unique properties, such as having a graph that is symmetrical about the origin, having a period (interval of repetition), and having a sine or cosine term in their equation. They can also be identified by their specific shape, which is typically a curve that starts at the origin, crosses through the x-axis, and repeats itself in a mirror image on the other side of the origin.

What is the difference between odd, periodic functions and even, periodic functions?

The main difference between odd, periodic functions and even, periodic functions is their symmetry. While odd, periodic functions are symmetric about the origin, even, periodic functions are symmetric about the y-axis. Additionally, odd, periodic functions have a sine term in their equation, while even, periodic functions have a cosine term.

How are odd, periodic functions used in real life?

Odd, periodic functions can be used to model various real-life phenomena, such as sound waves, light waves, and the movement of pendulums. They can also be used in fields such as engineering, physics, and statistics, to analyze and predict patterns and behaviors of different systems and processes.

What are some examples of odd, periodic functions?

Some common examples of odd, periodic functions include the sine function (y = sin(x)), tangent function (y = tan(x)), and cosecant function (y = csc(x)). Other examples include the sawtooth wave, square wave, and triangle wave functions, which are commonly used in signal processing and electronics.

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