Suppose $\phi: \mathbb{R} \rightarrow \mathbb{R}$ . Assuming the necessary conditions, then its
- Full Fourier series on $[-l, l]$ is $$\phi(x) = \frac{1}{2}A_0 + \sum_{n=1}^\infty A_n\cos\big(\frac{n\pi x}{l}\big) + B_n\sin\big(\frac{n\pi x}{l}\big)$$ where $$A_n = \frac{1}{l}\int_{-l}^l \phi(x) \cos\big(\frac{n\pi x}{l}\big)dx, \quad B_n = \frac{1}{l}\int_{-l}^l \phi(x) \sin\big(\frac{n\pi x}{l}\big)dx$$
- Fourier Cosine series on $[0, l]$ is $$\phi(x) = \frac{1}{2}A_0 + \sum_{n=1}^\infty A_n\cos\big(\frac{n\pi x}{l}\big)$$ where $$A_n = \frac{2}{l}\int_{0}^l \phi(x) \cos\big(\frac{n\pi x}{l}\big)dx$$
- Fourier Sine series on $[0,l]$ is $$\phi(x) = \sum_{n=1}^\infty B_n\sin\big(\frac{n\pi x}{l}\big)$$ where $$B_n = \frac{2}{l}\int_{0}^l \phi(x) \sin\big(\frac{n\pi x}{l}\big)dx.$$
Based on my readings I was under the impression that formulas transfer practically identically to polar coordinates. That is,
- Full Fourier series is $$\phi(\theta) = \frac{1}{2}A_0 + \sum_{n=1}^\infty A_n\cos\big(\frac{n\pi \theta}{l}\big) + B_n\sin\big(\frac{n\pi \theta}{l}\big)$$ where $$A_n = \frac{1}{\pi}\int_{0}^{2\pi} \phi(\theta) \cos\big(\frac{n\pi \theta}{l}\big)d\theta, \quad B_n = \frac{1}{\pi}\int_{0}^{2\pi} \phi(\theta) \sin\big(\frac{n\pi \theta}{l}\big)d\theta$$
- Fourier Cosine series is $$\phi(\theta) = \frac{1}{2}A_0 + \sum_{n=1}^\infty A_n\cos\big(\frac{n\pi \theta}{l}\big)$$ where $$A_n = \frac{2}{\pi}\int_{0}^{2\pi} \phi(\theta) \cos\big(\frac{n\pi \theta}{l}\big)d\theta$$
- Fourier Sine series is $$\phi(\theta) = \sum_{n=1}^\infty B_n\sin\big(\frac{n\pi \theta}{l}\big)$$ where $$B_n = \frac{2}{\pi}\int_{0}^{2\pi} \phi(\theta) \sin\big(\frac{n\pi \theta}{l}\big)d\theta.$$
However in my experience of using Fourier series to solve PDEs using separation of variables, this does not always seem to be the case (for example, page 167 in Strauss's text Partial Differential Equations and these notes from Brown disagree on the coefficient when solving Laplace's equation on the exterior of a circle, this Wikipedia article also has the coefficient of a full series as $1/\pi$, does Strauss have a typo?). I sometimes see that the scalar in front of the integral defining the coefficients to be $2/\pi$ and other times $1/\pi$. Which one is correct, or is it done on a case by case basis?