PHZ3113–Introduction to Theoretical PhysicsFall 2008Problem Set 13Oct. 29, 2008Due: Friday, Nov. 7, 2008Reading: Boas Ch. 21. A plane wave of light of angular frequency ω is represented byeiω(t−nx/c), (1)where t is time, x is distance, c is the speed of light, and n is the index ofrefraction. In a certain medium, it is found that n is a complex quantity,n = n0+ in00, where n0and n00are real numbers. Find the real part of theexpression above. What is the qualitative effect of n00on the wave? What doesn00correspond to physically?2. For the following pairs of numbers z1and z2, give their polar form; their complexconjugates, their moduli (magnitudes), product, and the quotient z1/z2:z1=1 + i√2; z2=√3 − i (2)z1=3 + 4i3 − 4i; z2=·1 + 2i1 − 3i¸2(3)3. Show using de Moivre’s formula that(a) cos nθ = cosnθ −¡n2¢cosn−2θ sin2θ +¡n4¢cosn−4θ sin4θ . . .(b) sin nθ =¡n1¢cosn−1θ sin θ −¡n3¢cosn−3θ sin3θ +¡n5¢cosn−5θ sin5θ . . .where the¡nm¢are binomial coefficients.4. (a) Write out the power series for sin z, cos z, sinh z, cosh z.(b) Assume that these functions are defined by their power series. Show thati sin z = sinh iz ; sin iz = i sinh z (4)cos z = cosh iz ; cos iz = cosh z. (5)(c) Verify, using the power series, that cosh z = (ez+ e−z)/2, i.e. that theusual relationship holds in the complex plane.1Figure 1: RLC circuit.5. (a) Find the total effective impedance of the combination shown, to be placedin a circuit at the two end wires.(b) Find ω at resonance (at resonance, ImZ =