1. Suppose
real. Then since
we obtain
all the inverse tangents lying between
and
. In particular, if we suppose
,
, and equate the real and imaginary parts, we obtain
2. Verify the formulae of Ex. 1 when , , . [Of course when is a positive integer the series is finite.]
3. Prove that if then
[Take
in the last two formulae of Ex. 1.]
4. Prove that if then for all real values of . [These results follow at once from the equations
5. We proved (Ex. LXXXI. 6), by direct multiplication of series, that , where , satisfies the functional equation Deduce, by an argument similar to that of § 216, and without assuming the general result above, that if is real and rational then
6. If and are real, and , then