Let f be a function with 2pi period that is piecewise-continous in R.
Let's define a function g such as g(x)=-f(-x) .
Find the connection between the fourier coeffeicients of f(x) and the fourier coefficieients of g(x)...
Thanks a lot!
Let f be a function with 2pi period that is piecewise-continous in R.
Let's define a function g such as g(x)=-f(-x) .
Find the connection between the fourier coeffeicients of f(x) and the fourier coefficieients of g(x)...
Thanks a lot!
Use the odd/even properties of $\displaystyle \sin$ and $\displaystyle \cos $ and the occaisional change of variable $\displaystyle x'=-x$:
$\displaystyle \int_{-\pi}^{\pi} g(x) \sin(x)\;dx=\int_{-\pi}^{\pi} -f(-x) \sin(x)\;dx= $ $\displaystyle \int_{\pi}^{-\pi} f(x') \sin(-x')\;dx'=\int_{-\pi}^{\pi} f(x') \sin(x')\;dx'$
and something similar-ish for $\displaystyle \cos$ will allow you to deduce the relationship between the coeffients for $\displaystyle f$ and $\displaystyle g$.
CB
I don't want you to tell me the answer ofcourse... But from what you've done we can't deduce anything about the needed relations... You've only showed a relation between the coefficients of f(-x) and g(x) and I've managed to get there on my own... The problem is to continue from there...
Hope you will be able to help me
Thanks a lot anyway
Actually I think I have indicated how to show the relation between the coefficients of $\displaystyle f(x)$ and $\displaystyle g(x)$ not $\displaystyle f(-x)$ and $\displaystyle g(x).$
$\displaystyle
\int_{-\pi}^{\pi} g(x) \sin(n x)\;dx=\int_{-\pi}^{\pi} -f(-x) \sin(n x)\;dx=
$$\displaystyle \int_{\pi}^{-\pi} f(x') \sin(- n x')\;dx'=\int_{-\pi}^{\pi} f(x') \sin(n x')\;dx'= $ $\displaystyle \int_{-\pi}^{\pi} f(x) \sin(n x)\;dx
$
since as $\displaystyle x'$ is a dummy variable of integration and can be replaced by anything.
CB