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I'm attempting to derive an analytical formula for the following sum:

\begin{align} \sum_{j=1}^k \sin(\theta j)\sqrt{\sin^2(\theta j)+a}, \end{align}

where $a \geq 0$. If anyone has any insights on how to find such a formula, or if it is not possible to express it analytically, please share your thoughts. Thank you in advance !

Tian Vlašić
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Joe
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  • Presumably you mean closed formula when you say analytic formula, because this is an analytic formula. – Thomas Andrews Jun 25 '23 at 16:54
  • If $b=1+a$ then $$\sqrt{\sin^2 \alpha+a}=\sqrt b\cdot \sqrt{1-\cos^2\alpha/b}$$ So you might be able to use the power series of $\sqrt{1-z}$ for $|z|<1$ and use that to get the Fourier series for $f(\theta)=\sin\theta \sqrt{\sin^2\theta+a}$ and thus implicitly for $f_i(\theta)=f(i\theta).$ But it seems unlikely that will give you anything more useful than the given sum. – Thomas Andrews Jun 25 '23 at 17:05
  • @ThomasAndrews Thank you for your answer ! Yes, by analytical formula I mean a closed formula. I already tried what you said but yeah it's not so useful ! Thanks anyway – Joe Jun 25 '23 at 18:44

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