Plane trigonometry, by S.L. Loney.

[Exs. LVIII.]

COMPLEX INDICES.

393

8. If aat'i=(x+yi)P+!i, principal values only being considered,
prove that
a = plog, (2 + y2) - q tan- log e,
a=~p Xlog,p e,
and that log, (x2 + y2) = 2 p + Pq.
9. Prove that the real part of the principal value of (i) log(l+i) is
_7r2
e 8 s (-1og 2).
10. Prove that the principal value of (a +ib)a+i' is wholly real or
wholly imaginary according as
1 b
2 P log (a2 + b2) + a tan-1
2 a
is an even or an odd multiple of.
2'
11. Prove that the general value of
(1 + i tan a)-i
is ea+22n7 [cos {log cos a} +i sin {log cos a}].
12. If (a+x+iy)A+i=X+iY,
\a-x- iy
prove that one of the values of
tan-l' is Xtan-1 2ay o () + 2 log $
a V2- ^2- y 2 (a- X)2 + y2
13. Prove that Log^ /- = 4 + 1 -where m and n are any integers.
14. Prove that the general value of Log4 (- 2) is
(log 2)2 + m. (2n + 1) 2 (2n + 1 - ) r log 2
2 (log 2)2 + 2m2r2 + 2 (log 2)2 + 2rm27r2
Explain the fallacies in the following arguments:
15. For all integral values of n we have
e2ni = cos 2nr + i sin 2nr = 1,
so that e2i =e4r =7i.......