Half-Range Fourier Series/Identity Function

Theorem
Let $\lambda \in \R_{>0}$ be a strictly positive real number.

Let $\map f x: \openint 0 \lambda \to \R$ be the identity function on the open real interval $\openint 0 \lambda$:
 * $\forall x \in \openint 0 \lambda: \map f x = x$

The half-range Fourier series of $f$ over $\openint 0 \lambda$ can be given in the following forms: