Half-Range Fourier Sine Series/Sine of Non-Integer Multiple of x over 0 to Pi

Theorem
Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer.

Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:


 * $\map f x = \sin \lambda x$

Then its half-range Fourier sine series can be expressed as:

Proof
By definition of half-range Fourier sine series:


 * $\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \sin n x$

where for all $n \in \Z_{> 0}$:
 * $a_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$

Because $\lambda \notin \Z$ we have that $\lambda \ne n$ for all $n$.

Thus for $n > 0$:

Finally:

Also presented as
Some sources present this result as: