information:highlighting, stylistic:tone, linguistic:voice_change, structural:expansion
Exponential form coefficients

Another applicable identity is Euler's formula:

(Note: the ∗ denotes complex conjugation.)

Substituting this into Eq.1 and comparison with Eq.3 ultimately reveals:

Conversely:

A 0 = C 0 A n = C n + C − n for n > 0 B n = i (C n − C − n) for n > 0 {\displaystyle {\begin{aligned}A_{0}&=C_{0}&\\A_{n}&=C_{n}+C_{-n}\qquad &{\textrm {for}}~n>0\\B_{n}&=i(C_{n}-C_{-n})\qquad &{\textrm {for}}~n>0\end{aligned}}}

Substituting Eq.5 into Eq.6 also reveals: