Equivalence of Definitions of Trigonometric Series

Proof
Let $\map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$.

Set:

Then:

Let $b_0 := 0$.

Then let:
 * $\forall n \in \N: c_n = \dfrac {a_n - i b_n} 2$

Thus it follows that:


 * $c_0 = \dfrac {a_0 - 0 \times i} 2 = \dfrac {a_0} 2$

Then let:
 * $c_{-n} = \dfrac {a_n + i b_n} 2$

from which it follows that:


 * $c_{-n} = \overline {\paren {\dfrac {a_n - i b_n} 2} }$

and again:


 * $c_0 = \dfrac {a_0 + 0 \times i} 2 = \dfrac {a_0} 2$

Then it is noted that:
 * $e^{i \times 0 \times x} = e^0 = 1$

and it follows that:

where it is noted that $c_{-n} = \overline {c_n}$ as required.