# Thread: Lebesgue integral of Odd Function.

1. ## Lebesgue integral of Odd Function.

Suppose i have f is an odd function.
How to show that $\displaystyle \int_{[-\pi,\pi]} f(x)$ =0 ?

2. ## Re: Lebesgue integral of Odd Function.

Originally Posted by younhock
Suppose i have f is an odd function.
How to show that $\displaystyle \int_{[-\pi,\pi]} f(x)$ =0 ?
This is true if f(x) has no singularities in $\displaystyle [-\pi,\pi]$... consider for example $\displaystyle f(x)=\frac{1}{x}$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. ## Re: Lebesgue integral of Odd Function.

Originally Posted by chisigma
This is true if f(x) has no singularities in $\displaystyle [-\pi,\pi]$... consider for example $\displaystyle f(x)=\frac{1}{x}$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
Is there any way to show it? Thank you.

4. ## Re: Lebesgue integral of Odd Function.

An odd function that is not the null function is positive in a part and negative in a different part of the definition interval. Let's define $\displaystyle f^{+} (x)= f(x)$ where $\displaystyle f(x)>0$ and $\displaystyle f^{+} (x)= 0$ elsewhere and $\displaystyle f^{-} (x)= -f(x)$ where $\displaystyle f(x)<0$ and $\displaystyle f^{-} (x)= 0$ elsewhere. The Lebesgue integral of f(x) is defined as...

$\displaystyle \int_{\mu} f(x) dx = \int_{\mu} f^{+}(x) dx - \int_{\mu} f^{-}(x) dx$ (1)

Now the integral (1) exists only if both the integrals in (1) exist and that is not true [for example...] for $\displaystyle f(x)= \frac{1}{x}$ in $\displaystyle [- \pi, +\pi]$...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

5. ## Re: Lebesgue integral of Odd Function.

Well, $\displaystyle \frac{1}{x}$ isn't Lebesgue integrable on $\displaystyle [-\pi,\pi]$, so it's a moot point, no? The basic idea is that change of variables still works for Lebsgue integrals. What if you let $\displaystyle x=-z$?

6. ## Re: Lebesgue integral of Odd Function.

Originally Posted by Drexel28
Well, $\displaystyle \frac{1}{x}$ isn't Lebesgue integrable on $\displaystyle [-\pi,\pi]$, so it's a moot point, no? The basic idea is that change of variables still works for Lebsgue integrals. What if you let $\displaystyle x=-z$?
I'm very sorry but I'm afraid not to have fully undestood Your reply... can You explain a little more, please?...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

7. ## Re: Lebesgue integral of Odd Function.

Originally Posted by chisigma
I'm very sorry but I'm afraid not to have fully undestood Your reply... can You explain a little more, please?...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$
Which part of it? The fact that $\displaystyle \frac{1}{x}$ isn't Lebesgue integrable, or how the change of variables works?

8. ## Re: Lebesgue integral of Odd Function.

Originally Posted by Drexel28
Which part of it? The fact that $\displaystyle \frac{1}{x}$ isn't Lebesgue integrable, or how the change of variables works?
The original question was to demonstrate that if f(x) is and odd function, then ...

$\displaystyle \int_{-\pi}^{+\pi} f(x) dx =0$ (1)

... where the integral is 'Lebesgue Integral'. My replay has been [symply...] that an odd function isn't necessarly Lebesgue integrable and the example $\displaystyle f(x)= \frac{1}{x}$ has been supplied. Of course if we add the condition that f(x) must be 'regular', then the demonstration of (1) is very comfortable...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$

9. ## Re: Lebesgue integral of Odd Function.

Originally Posted by chisigma
The original question was to demonstrate that if f(x) is and odd function, then ...

$\displaystyle \int_{-\pi}^{+\pi} f(x) dx =0$ (1)

... where the integral is 'Lebesgue Integral'. My replay has been [symply...] that an odd function isn't necessarly Lebesgue integrable and the example $\displaystyle f(x)= \frac{1}{x}$ has been supplied. Of course if we add the condition that f(x) must be 'regular', then the demonstration of (1) is very comfortable...

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$
Clearly we are to assume that the function is integrable, otherwise the integral doesn't make sense! That's like saying "Prove that the limit of a sequence in $\displaystyle [0,\infty)$ is in $\displaystyle [0,\infty)$ is false since $\displaystyle \lim n$ is infinite--it's clear by wording that we are to assume the sequence converges!

10. ## Re: Lebesgue integral of Odd Function.

Maybe I should add one more condition to the question since it causes confusion. Now f(x) is lebesgue integrable and odd. how to show the integral value is zero?

11. ## Re: Lebesgue integral of Odd Function.

Originally Posted by younhock
Maybe I should add one more condition to the question since it causes confusion. Now f(x) is lebesgue integrable and odd. how to show the integral value is zero?
... if f(x) is odd then $\displaystyle f(x)=-f(-x)$ so that is...

$\displaystyle \int_{-\pi}^{0} f(x)\ dx = - \int_{0}^{\pi} f(x)\ dx \implies \int_{-\pi}^{0} f(x)\ dx + \int_{0}^{\pi} f(x)\ dx = 0$

Marry Christmas from Serbia

$\displaystyle \chi$ $\displaystyle \sigma$