Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$

Question

Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$ I tried WA; it does not return a function.

Answer 1

Rewrite the series as

$$\Re{\left [ \sum_{n=0}^{\infty} \frac{\left ( e^{i x}\right )^{2 n+1}}{2 n+1} \right]} = \Re{[\text{arctanh}{(e^{i x})}]} = \frac12 \Re{\left[\log{\left(\frac{1+e^{i x}}{1-e^{i x}}\right)}\right]}$$

With some manipulation,noting that $\log{z} = \log{|z|} + i 2 \pi \arg{z}$, we find that

$$\sum_{n=0}^{\infty} \frac{\cos{(2 n+1) x}}{2 n+1} = \frac12 \log{\left |\cot{\frac{x}{2}}\right |}$$