PX153 - J7 - general interval

f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos (\frac{n π x}{L}) + b_{n} \sin (\frac{n π x}{L})) $ $ f o r $ x \in [L, L) $ - f r o m [[c o n t e n t / 011 / P X 153 - m a t h e m a t i c s f o r p h y s i c i s t s / t e r m 2 / P X 153 - J - f o u r i e r s e r i e s / P X 153 - J 3 - p r o o f s a n d d e r i v a t i o n s ∥ p r o o f s a n d d e r i v a t i o n s]] : $ $ \begin{aligned} a_{n} & = \frac{1}{L} \int_{- L}^{L} f (x) \cos (\frac{n π x}{L}) d x \\ b_{n} & = \frac{1}{L} \int_{- L}^{L} f (x) \sin (\frac{n π x}{L}) d x \\ a_{0} & = \frac{1}{L} \int_{- L}^{L} f (x) d x \end{aligned}

\int_{- L}^{L} f (x) \cos [\frac{m π x}{L}] d x