Physics 263: Problem Set #10

1. (BTM 6.2.5) Write out sin(z) in terms of exponentials and then apply e^iy = cos(y) + i*sin(y) and do some rearrangement. You'll have to use cos²x + sin²x = 1 and the corresponding relation for hyperbolic functions. If you solve this a different way, you may find the trig identities cos(2x) = 1 - 2 sin²x and cosh(2x) = 1 + 2 sinh²x helpful. Given your answers, under what conditions can sin(z) vanish (considering any complex z)?
2. (BTM 6.2.7) When you expand ln(1+z) in the exponent out to O(z³), treat that entire expansion as box and then expand e^box out to box³ as well. When you assemble it, throw out everything that is z⁴ or a greater power. But make sure you don't just keep 1+z in the exponent; this does not show that the z² and z³ terms cancel!
3. (BTM 6.2.11) Apply equation (6.2.41) with N=2 (or (6.2.30) and (6.2.32) directly). For the second part, you will find the half-angle formulas for cos(theta/2) and sin(theta/2) helpful in easily finding the answer in the back of the book.
4. (BTM 6.1.5) This is easy if you use the Cauchy-Riemann equations to check for analyticity (as defined in Chapter 6).
5. (BTM 6.3.2) Break the integral into the four pieces. On each leg, one of the two variables is constant, so you can substitute for it, leaving a total of four simple one-dimensional integrals. Remember that dz = dx + i*dy, so the vertical legs (where dy isn't zero) get a factor of i.
6. (BTM 6.4.6) You should first work through the integrals in section 6.4 (we probably won't do them in class). Then follow the hints given in the book. An important result to simplify your life: if f(z) = g(z)/h(z) and there is a pole at z=a, then the residue at a is g(a)/h'(a).