Identify

$f\left(x\right)=\sum _{k=-\mathrm{\infty}}^{\mathrm{\infty}}\frac{1}{{(x+k)}^{2}}$

Myah Fuller
2022-02-24
Answered

Identify

$f\left(x\right)=\sum _{k=-\mathrm{\infty}}^{\mathrm{\infty}}\frac{1}{{(x+k)}^{2}}$

You can still ask an expert for help

Athena Hussain

Answered 2022-02-25
Author has **5** answers

We can also use the well known summation formula

$\sum _{k\in \mathbb{Z}}f\left(k\right)=-\sum \left[\text{residues of}\text{}\pi \mathrm{cot}\left(\pi z\right)f\left(z\right)\text{}\text{at}\text{}f\left(z\right)textspoles\right]$

so we have to evaluate

$Re{s}_{z=-x}\frac{\pi \mathrm{cot}\left(\pi z\right)}{{(z+x)}^{2}}=\underset{z\to -x}{lim}\frac{d}{dz}\left(\pi \mathrm{cot}\left(\pi z\right)\right)=-{\pi}^{2}{\mathrm{csc}}^{2}\left(\pi x\right)$

hence

$\sum _{k\in \mathbb{Z}}\frac{1}{{(k+x)}^{2}}={\pi}^{2}{\mathrm{csc}}^{2}\left(\pi x\right)$

obviously if x is not an integer.

so we have to evaluate

hence

obviously if x is not an integer.

asked 2022-07-14

I am trying to implicitly differentiate

$\mathrm{sin}(x/y)=1/2$

The solution manual says

Step 1.

$\mathrm{cos}(x/y)\cdot \frac{y-x\frac{dy}{dx}}{{y}^{2}}=0$

But I don't understand how they arrive at this next part:

Step 2.

$y-x\frac{dy}{dx}=0$

Is $\mathrm{cos}(x/y)={y}^{2}$?

$\mathrm{sin}(x/y)=1/2$

The solution manual says

Step 1.

$\mathrm{cos}(x/y)\cdot \frac{y-x\frac{dy}{dx}}{{y}^{2}}=0$

But I don't understand how they arrive at this next part:

Step 2.

$y-x\frac{dy}{dx}=0$

Is $\mathrm{cos}(x/y)={y}^{2}$?

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