Proving Periodicity of an Odd Function with Period p

In summary, the conversation discusses proving the periodicity of a function that is both periodic and odd with a given period. The solution involves using the property of definite integrals and applying it to different intervals to show that the function is indeed periodic.
  • #1
zorro
1,384
0

Homework Statement


If f be a periodic function as well as an odd function with period p and and x belongs to [-p/2, p/2]. Prove that
gif.latex?\int_{a}^{x}f(t)dt.gif
is periodic with period p.


The Attempt at a Solution



In the solution, there is a step which I did not understand-

?\int_{0}^{p}f%28t%29dt&space;=&space;\int_{\frac{-p}{2}+0}^{\frac{-p}{2}+p}f%28t%29dt.gif


I see no property of definite integrals here. Help needed.
 
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  • #2
Hi,
this step is connected with the fact that the function f is periodic:

f(t+T)=f(t+T)

so the integral of this function in an interval [a,b] is equal to the integral in [a+T,b+T].
 
  • #3
But -p/2 is not a period of the function.
 
  • #4
[tex]\int_{0}^{p}f(t)dt=\int_{0}^{\frac{p}{2}}f(t)dt+\int_{\frac{p}{2}}^{p}f(t)dt[/tex]

then just apply the previous property at the second integral

[tex]\int_{\frac{p}{2}}^{p}f(t)dt=\int_{\frac{p}{2}-p}^{p-p}f(t)dt[/tex]

I hope that it is correct :blushing:
 
  • #5
How does that help? Your last integral doesnot lead to the step in the question.
 
  • #6
[tex]
\int_{0}^{p}f(t)dt=\int_{0}^{\frac{p}{2}}f(t)dt+\int_{\frac{p}{2}}^{p}f(t)dt
[/tex]

[tex]
\int_{\frac{p}{2}}^{p}f(t)dt=\int_{-\frac{p}{2}}^{0}f(t)dt
[/tex]

substituting in the first equation:

[tex]
\int_{0}^{\frac{p}{2}}f(t)dt+\int_{-\frac{p}{2}}^{0}f(t)=\int_{-\frac{p}{2}}^{\frac{p}{2}}f(t)dt
[/tex]
 
  • #7
Ah I was dumb there. Thank you very much for you help :smile:
 

1. What is an odd function?

An odd function is a mathematical function that satisfies the property of being symmetric about the origin. This means that if you reflect the graph of the function across the origin, it will remain unchanged. In other words, an odd function has rotational symmetry of 180 degrees around the origin.

2. What is the period of a function?

The period of a function is the distance, or interval, along the x-axis over which the function repeats itself. In other words, it is the smallest positive value of x for which the function f(x) = f(x + p), where p is the period. For example, the period of the function sin(x) is 2π because sin(x) = sin(x + 2π).

3. How do you prove periodicity of an odd function with period p?

To prove that an odd function is periodic with period p, you must show that f(x + p) = -f(x) for all values of x. This means that if you shift the function by its period p, the resulting function will be the negative of the original function. This can be done algebraically by substituting (x + p) for x in the function and simplifying to show that f(x + p) = -f(x).

4. Can an odd function have a period other than p?

No, an odd function can only have a period of p. This is because the property of being odd requires the function to be symmetric about the origin, and any other period would result in the function not being symmetric.

5. What is the relationship between the period and the graph of an odd function?

The period of an odd function is directly related to the shape of its graph. If the period is larger, the graph will be more spread out and have a lower frequency. If the period is smaller, the graph will be more condensed and have a higher frequency. Additionally, the graph will always be symmetric about the origin, regardless of the value of the period.

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