Adobe Photoshop CC 2015.1.2 (20160113.r.355) Cracked – AppzDam Setup Free

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Adobe Photoshop CC 2015.1.2 (20160113.r.355) Cracked – AppzDam Setup Free

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Using divisor in cosine series

I have a question which concerns calculating the series of cosine.
$$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^s}$$
I know how to calculate it if $x \in \mathbb{R}$. But now I’m interested in calculating it with $x \in \mathbb{C}$ (for $Re(x)>0$) in which case I know that $\frac{\cos(nx)}{n^s}
eq \frac{\cos(x)}{n^s}$ and $\frac{e^{i nx}}{n^s}
eq \frac{e^{ix}}{n^s}$.
How can I calculate it with $x \in \mathbb{C}$? If possible, I would like the result to be valid for $Re(s)>\frac{1}{2}$ and I would like to calculate it to higher degrees of accuracy.

A:

Given that you are looking for the best accuracy, you should rather consider the power series of $e^{2\pi ix}$ and the cosine series. The power series expansion of $e^{2\pi ix}$ is
$$e^{2\pi ix}=\sum_{n\geq0}\frac{(2\pi ix)^n}{n!}=\sum_{n\geq0}\frac{(2\pi )^n}{n!}x^n=\sum_{n\geq0}\frac{1}{n!}x^n.\tag1$$
Similarly the power series expansion of $e^{2\pi ix}/

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