# Homework #6

Due Friday of Week 7 at 5pm.

*Functions on the complex plane*

**1.** Butkov 2.34 a) - f)

Purpose of question: You already have good intuition about functions of a single real variable, for example, you know the function 1/x blows up at x = 0 and exp(x) blows up as x → ∞. It takes some practice to develop a similar intuition for functions of a single complex variable. The Argand plane can be a minefield of singularities/divergences.

*Fourier Series*

**2.** Butkov 4.3

**3.** Butkov 4.4

**4.** Butkov 4.12

*Fourier Transforms*

**5.** Find the Fourier transform representation of 1/(*x*^4 + *a*^4) and show that Δ*x*Δ*k* satisfies the “uncertainty principle for a wavepacket”. It is not necessary to calculate Δ*x* and Δ*k* precisely, you can use an approximation such as full-width at half max.