Proving Periodicity of an Odd Function with Period p

[zorro]

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

is periodic with period p.

The Attempt at a Solution

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

I see no property of definite integrals here. Help needed.

 

[accatagliato]

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].

 

[zorro]

But -p/2 is not a period of the function.

 

[accatagliato]

$$\int_{0}^{p}f(t)dt=\int_{0}^{\frac{p}{2}}f(t)dt+\int_{\frac{p}{2}}^{p}f(t)dt$$

then just apply the previous property at the second integral

$$\int_{\frac{p}{2}}^{p}f(t)dt=\int_{\frac{p}{2}-p}^{p-p}f(t)dt$$

I hope that it is correct

 

[zorro]

How does that help? Your last integral doesnot lead to the step in the question.

 

[accatagliato]

$$\int_{0}^{p}f(t)dt=\int_{0}^{\frac{p}{2}}f(t)dt+\int_{\frac{p}{2}}^{p}f(t)dt$$

$$\int_{\frac{p}{2}}^{p}f(t)dt=\int_{-\frac{p}{2}}^{0}f(t)dt$$

substituting in the first equation:

$$\int_{0}^{\frac{p}{2}}f(t)dt+\int_{-\frac{p}{2}}^{0}f(t)=\int_{-\frac{p}{2}}^{\frac{p}{2}}f(t)dt$$

 

[zorro]

Ah I was dumb there. Thank you very much for you help