Science:Math Exam Resources/Courses/MATH307/December 2010/Question 06 (d)/Solution 1

From part (a), we figured out that $c_{n}=-{\frac {1}{2\pi in}}$. The modulus of this complex number, $\left|c_{n}\right|$, is the amplitude of oscillation with frequency $\omega _{n}=n$ since L is equal to 1. We have that

$\left|c_{n}\right|=\left|-{\frac {1}{2\pi in}}\right|={\frac {1}{2\pi n}}\left|{\frac {-1}{i}}\right|={\frac {1}{2\pi n}}$

for $n\neq 0$. When $n=0$, the amplitude is equal to $\left|c_{0}\right|$ which is $c_{0}={\frac {1}{2}}$ from part (a). Therefore the points for a frequency amplitude plot are $\displaystyle (n,|c_{n}|)$ given by

${\begin{aligned}(0,&1/2)\\(1,&1/2\pi )\\(2,&1/4\pi )\\&\vdots \\(n,&1/2n\pi )\end{aligned}}$