What function does this trigonometric series represent?

Question

I wonder what known function does this trigonometric series represent?

$$ f(x)=\sum_{k=1}^\infty \frac{1}{k^{3/2}}\sin{(kx)} $$ with $$ -\pi\le x\le \pi $$ and $$ f(x)=f(x+2\pi) $$

And how is it obtained?

Here is a plot of two periods using 100 terms:

Thank you!

Answer 1

We may formally express the series in terms of the polylogarithm defined as $\displaystyle\text{Li}_p(z):=\sum_{k=1}^\infty \dfrac{z^k}{k^{p}}$. Since $\sin(kx)=\text{Im}\,(e^{ikx})$, this series is the imaginary part of the summation $\displaystyle\sum_{k=1}^\infty \dfrac{e^{ikx}}{k^{3/2}}$. Therefore we may write $\boxed{f(x)=\text{Im}\,\left[\text{Li}_{3/2}(e^{i x})\right]}$.

However, this formal approach is actually a bit careless: the definition of the polylog cited above only converges for $|z|<1$, which is equivalent to requiring $x$ to have positive imaginary part. I'll see if I can clear up this subtlety.

Answer 2

It looks like it could be a function of the form $$y=A\sin\left[x-\left(\frac{\pi}2-\alpha\right)y\right]$$ where $A$ is the amplitude and $\alpha$ ($0<\alpha<\frac\pi2$) is the angle where the function is maximum.

See my response to another related question here:

Skewed Trigonometric Function