- #1

Zaare

- 54

- 0

[tex]

\hat f\left( n \right) = - \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {f\left( {x + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} \right)e^{ - inx} dx}

[/tex]

where [tex]\hat f\left( n \right)[/tex] is the Fourier coefficient and [tex]f(x)[/tex] is a [tex]2\pi[/tex]-periodic and Riemann integrable on [tex][\pi,-\pi][/tex].

This is what I've done:

I use the formulae for the Fourier coefficient of a [tex]2\pi[/tex]-periodic function

[tex]

\hat f\left( n \right) = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {f\left( x \right)e^{ - inx} dx}

[/tex]

and a simple change of variable

[tex]

\hat f\left( n \right) = \frac{1}{{2\pi }}\int\limits_{ - \pi + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}}^{\pi + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} \right)e^{ - in\left( {x + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} \right)} dx} = \frac{1}{{2\pi }}\int\limits_{ - \pi + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}}^{\pi + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} \right)e^{ - inx} e^{ - i\pi } dx} = - \frac{1}{{2\pi }}\int\limits_{ - \pi + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}}^{\pi + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} {f\left( {x + {\pi \mathord{\left/

{\vphantom {\pi n}} \right.

\kern-\nulldelimiterspace} n}} \right)e^{ - inx} dx}

[/tex].

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 [tex]2\pi[/tex] apart?