Calculating Fourier Coefficient of 2π-Periodic Function

[Zaare]

I'm supposed to show
<br /> \hat f\left( n \right) = - \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {f\left( {x + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} \right)e^{ - inx} dx}<br />
where \hat f\left( n \right) is the Fourier coefficient and f(x) is a 2\pi-periodic and Riemann integrable on [\pi,-\pi].

This is what I've done:
I use the formulae for the Fourier coefficient of a 2\pi-periodic function
<br /> \hat f\left( n \right) = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {f\left( x \right)e^{ - inx} dx}<br />
and a simple change of variable
<br /> \hat f\left( n \right) = \frac{1}{{2\pi }}\int\limits_{ - \pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}}^{\pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} \right)e^{ - in\left( {x + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} \right)} dx} = \frac{1}{{2\pi }}\int\limits_{ - \pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}}^{\pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} \right)e^{ - inx} e^{ - i\pi } dx} = - \frac{1}{{2\pi }}\int\limits_{ - \pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}}^{\pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} \right)e^{ - inx} dx}<br />.

Everything seems to agree except for the limits of the integral.
1) If I have done some mistake, I'd appreciate it someon would point it out.
2) If I haven't done any mistakes, what's the reasoning behind this? Don't the limits matter as longs as they are 2\pi apart?

 

[dextercioby]

You'll have to prove that

\int_{-\pi+\frac{\pi}{n}}^{-\pi} f\left(x+\frac{\pi}{n}\right) \ dx =-\int_{\pi}^{\pi+\frac{\pi}{n}} f\left(x+\frac{\pi}{n}\right) \ dx

Daniel.

 

[Zaare]

Ok, here's my reasoning.
The integrand

<br /> f(x+\frac{\pi}{n})e^{-inx}<br />

is 2\pi-periodic. Hence a change from x+\frac{\pi}{n} to x+\frac{\pi}{n}+2\pi leaves the integrand unchanged and I get

<br /> \int\limits_{ - \pi }^{ - \pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} \right)e^{ - inx} dx} = \int\limits_\pi ^{\pi + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/<br /> {\vphantom {\pi n}} \right.<br /> \kern-\nulldelimiterspace} n}} \right)e^{ - inx} dx}<br />

which allows me to change the limits of the original integral to -\pi and \pi without changing its value.
Am I right?