Plane trigonometry, by S.L. Loney.

EXPANSIONS.

417

Since 2 cos 0 = eli + e-~1,
we have
log (1 - 2a cos 0 + a2) = log [1 - a (e0i + e-i) + a2]
= log [(1 -aei) (1 - e-i)]
= log (1 - aeOi) + log (1 - ae-0i)
= - aeOi - a2e20i a3e3i -1 a4e4i..
2 4
- ae-0i - a2e-2-i 1 a3e-30i
2 3
= a [- i a + [es +~ e —20] 1- a3 [e30i + e-30i]
2 3
=-a. 2cos0- a. 2cos20 - a.2cos 30......
2 3
=- L2 acacos 2os 20+ a3 cos 30 +.........
The expansion of log(1 - ae0i) is legitimate, by Art.
336, if the modulus of - ae0 be less than unity.
Now - ae0i = a cos (7r + 0) + i sin (wr + 0),
so that its modulus is equal to a. Hence the above
expansion is legitimate provided that a is less than unity.
The expansion is also legitimate if a be equal to unity
provided that 0 do not equal an even multiple of tr.
It is also legitimate if a be equal to - 1 and 0 do not
equal an odd multiple of r.
358. Ex. Expand
-a2
1 - 2a cos 0 + a2
in a series of ascending powers of a.

L. T.

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