91Ó°ÊÓ

In optics, the following expression needs to be evaluated in calculating the intensity of light transmitted through a film after multiple reflections at the surfaces of the film:

${\left(\underset{n=0}{\overset{∞}{∑}}{r}^{2n}cosn\mathrm{θ}\right)}^{\mathbf{2}}\mathbf{+}{\left(\underset{n=0}{\overset{∞}{∑}}{r}^{2n}sin\mathrm{θ}\right)}^{\mathbf{2}}$

Show that this is equal to${\left|\underset{n=0}{\overset{∞}{∑}}{r}^{2n}{e}^{in\mathrm{θ}}\right|}^{\mathbf{2}}$and so evaluate it assuming |r| < 1 (r is the fraction of light reflected each time)

Find one or more values of each of the following complex expressions and compare with a computer solution.

${\left(\frac{1+i}{1-i}\right)}^{\mathbf{2718}}$

Express the following complex numbers in the x+iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

23. $\frac{{(1+i)}^{48}}{{(\sqrt{3}−i)}^{25}}$

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

$\frac{\mathbf{z}}{\stackrel{\mathbf{¯}}{\mathbf{z}}}$.

Find each of the following in rectangular form x + iyand check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{s}\mathit{i}\mathit{n}\left(\mathrm{Ï€}-i\mathrm{In}3\right)$.