3—Complex Algebra 663.52 Confirm the plot ofln(1+iy)following Eq. (3.15). Also do the corresponding plots forln(10+iy)
andln(100 +iy). And what do these graphs look like if you take the other branches of the logarithm,
with thei(θ+ 2nπ)?
3.53 Check that the results of Eq. (3.14) for cosines and for sines give the correct results for smallθ?
What aboutθ→ 2 π?
3.54 Finish the calculation leading to Eq. (3.13), thereby proving that the two indicated lines have the
same length and are perpendicular.
3.55 In the same spirit as Eq. (3.13) concerning squares drawn on the sides of an
arbitrary quadrilateral, start with an arbitrary triangle and draw equilateral triangles
on each side. Find the centroids of each of the equilateral triangles and connect them.
The result is an equilateral triangle. Recall: the centroid is one third the distance from
the base to the vertex. [This one requires more algebra than the one in the text.]
(Napoleon’s Theorem)