In my Fourier text book, there are the following exercises to prove. why do some of them have the same left side but have different right sides?

Question

In my Fourier text book, there are the following exercises to prove.why do some of them have the same left side but have different right sides? The demand of these question is to prove these equations by Parseval theorem. It is best that you can give a detailed answer.

Answer 1

I believe these formulas were accompanied by text that explains the nature of $f$ and its relation to $a_b,b_n,c_n$. For example:

• (a) concerns the exponential form of Fourier series, $\sum_{n=-\infty}^\infty c_n e^{inx}$
• (b) concerns the trigonometric Fourier series, $\frac{a_0}{2}+\sum_{n=1}^\infty (a_n \cos nx+b_n\sin nx)$
• (c) is about the trigonometric Fourier series of an odd function $f$. Such a function has only sines in its expansion, no cosines: $\sum_{n=1}^\infty b_n\sin nx$
• Similarly, (d) is about the trigonometric Fourier series of an even function $f$. Such a function has only cosines in its expansion: $\frac{a_0}{2}+\sum_{n=1}^\infty a_n \cos nx$
• Finally, (e) and the last item named "(d)" concern a function defined on an interval $[0,L]$. Such a function can be extended to $[-L,L]$ either as odd or an even function. Then one uses (c) or (d), as appropriate.

You will find a detailed explanation of sine and cosine series here.