3—Complex Algebra 63cos(
kr 0 −ωt
)
+ cos(
k(r 0 −dsinθ)−ωt
)
+ cos(
k(r 0 − 2 dsinθ)−ωt
)
+
...+ cos
(
k(r 0 −Ndsinθ)−ωt
)
Express this as the real part of complex exponentials and sum the finite series. Show that the resulting
wave is
sin
( 1
2 (N+ 1)kdsinθ
)
sin( 1
2 kdsinθ
) cos(
k(r 0 −^12 Ndsinθ)−ωt
)
Interpret this result as a wave that appears to be coming from some particular point (where?) and with
an intensity pattern that varies strongly withθ.
3.25 (a) If the coefficients in a quadratic equation are real, show that ifzis a complex root of the
equation then so isz*. If you do this by reference to the quadratic formula, you’d better find another
way too, because the second part of this problem is
(b) Generalize this to the roots of an arbitrary polynomial with real coefficients.
3.26 You can represent the motion of a particle in two dimensions by using a time-dependent complex
number withz=x+iy=reiθshowing its rectangular or polar coordinates. Assume thatrandθare
functions of time and differentiatereiθto get the velocity. Differentiate it again to get the acceleration.
You can interpreteiθas the unit vector along the radius andieiθas the unit vector perpendicular to
the radius and pointing in the direction of increasing theta. Show that
d^2 z
dt^2
=eiθ
[
d^2 r
dt^2
−r
(
dθ
dt
) 2 ]
+ieiθ
[
r
d^2 θ
dt^2
+ 2
dr
dt
dθ
dt
]
(3.17)
and translate this into the usual language of components of vectors, getting the radial (rˆ) component
of acceleration and the angular component of acceleration as in section8.9.
3.27 Use the results of the preceding problem, and examine the case of a particle moving directly away
from the origin. (a) What is its acceleration? (b) If instead, it is moving atr=constant, what is its
acceleration? (c) If instead,x=x 0 andy=v 0 t, what arer(t)andθ(t)? Now computed^2 z/dt^2 from
Eq. (3.17).
3.28 Was it really legitimate simply to substitutex+iyforθ 1 +θ 2 in Eq. (3.11) to getcos(x+iy)?
Verify the result by substituting the expressions forcosxand forcoshyas exponentials to see if you
can reconstruct the left-hand side.
3.29 The roots of the quadratic equationz^2 +bz+c= 0are functions of the parametersbandc.
For realbandcand for both casesc > 0 andc < 0 (say± 1 to be specific) plot the trajectories of
the roots in the complex plane asbvaries from−∞to+∞. You should find various combinations of
straight lines and arcs of circles.
3.30 In integral tables you can find the integrals for such functions as
∫dxeaxcosbx, or
∫
dxeaxsinbx
Show how easy it is to do these by doing both integrals at once. Do the first plusitimes the second
and then separate the real and imaginary parts.