Physics 263: Problem Set #8

1. (BTM 4.3.1) You are encouraged to use the MATLAB program from the 1094 session to check your answers. There are answers in the back of the book, although the answer for (vi) is not quite correct (it should be x > 1/2 and not |x| > 1/2). Also, give the range of x for (ii) and not just the radius of convergence.
2. (BTM 4.3.4) Nothing tricky here, just the (1+x)^p expansion. When working to order v^4, keep all v^4 terms but also drop all v^6 and higher terms. This may mean that you use the same expansion to higher order in one part of the E^2 = P^2 + m^2 equation than in another part.
3. (BTM 4.3.6) Note that we're expanding in k^2 sin^2(phi) and not expanding k separately. The answer is in the back for the specific case of alpha = pi/3.
4. (BTM 5.2.3) and (BTM 5.2.4) Remember that the real and imaginary parts of an equation are separately equal (so there are two independent equations in each real equation). I would multiply everything by (x+iy) to get it out of the denominator, rather than rationalizing by multiplying by (x-iy)/(x-iy). BTM 5.2.4 is straightforward; if you do z* before |z| you can use |z| = sqrt(z z*).
5. (BTM 5.2.5) What happens if you take the complex conjugate of the polynomial and set it to zero? How are the solutions to that equation related to the solutions of the original equation?