Answer Happy

a) The Fourier series for the sawtooth wave,
otherwise , has coefficients,
i. Explain why the coefficients 𝑎𝑚 are all zero.ii. Write down: a) the period and b) the Fourier series of thefunction S(t). You must explain why you picked the period for fullmarks.iii. Give an expression for the coefficients 𝑐𝑛 in the belowseries.
What is the Fourier transform of
For full marks, find the width of the Gaussian obtained byconvolution of normalised Gaussians with widths 𝑎1 and 𝑎2.
S(t)={tS(t±2π)​−π≤t<π otherwise ​
bm​=m2(−1)m​
S(t)=−∞∑∞​cn​eint
f(t)=e−∣t∣ is F(ω)=1+ω22​
h(t)=e−∣t∣/3
g(x)=a2π​1​e−2a2x2​
G(k)=F[g(x)](k)=e−2a2k2​