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      <text>23.6.2.4 Revision of the exponential form of a complex number</text>
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      <text><![CDATA[Rewriting in complex exponential form, we have
\[ a_n \cos n \theta + b_n \sin n\theta = \dfrac{a_n}{2} ( e^{in\theta} + e^{-in\theta} ) + \dfrac{b_n}{2i} ( e^{in\theta} - e^{-in\theta} ) \]
Now collecting the terms in \(e^{ in\theta}\) and in \(e^{ -in\theta}\)  and use the fact that \( \dfrac{1}{i} = -i \), we have
\[ \dfrac{1}{2} \left( a_n - ib_n \right) e^{ in\theta} + \dfrac{1}{2} \left( a_n + ib_n \right) e^{ -in\theta} \]
This can be expressed more concisely by defining 
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\( c_n = \dfrac{1}{2} (a_n - ib_n)  \) which has complex conjugate \( \displaystyle{c_n^{*}} = \dfrac{1}{2} (a_n + ib_n)  \) giving us
\[ a_n \cos n \theta + b_n \sin n\theta = c_n e^{i n \theta} + c_n^{*} e^{-i n \theta} \]]]></text>
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