3.
a) Working directly with exponentials, show that the magnitude of ˜z=Aeiωt is A.
Recall that the magnitude of a complex number ˜zis given by
|˜z|= [˜z∗˜z]1/2
Using the exponential form, as instructed in the problem, gives
|˜z|=A e−iωt A eiωt1/2=A21/2
Since it is assumed that Ais a real number when we write complex numbers in exponential form, we
can then simplify this to
|˜z|=A21/2=A
b) Show that the imaginary part of ˜zis given by (˜z−˜z∗)/2i. You do not need to use exponentials for (b).
It is probably easiest here to use the complex number representation ˜z=a+bi where ais the real part
and bis the imaginary part. Putting this into the expression above gives
(a+bi)−(a−bi)
2i=(a−a) + (bi +bi)
2i=2bi
2i=b
Given the definition of this form of complex number representation, bis the imaginary part of ˜
(z).