Intereting Posts

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Comparison of sequential compactness and limit point compactness.

$$\frac{d^n}{dx^n}\arctan(x)=\frac{(n-1)!}{(1+x^2)^{n/2}}\sin\left[n \arctan\left(x+\frac{\pi}{2}\right)\right]$$

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- homeomorphism from $\mathbb{R}^2 $ to open unit disk.
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HINT:

$$\dfrac{d(\arctan x)}{dx}=\dfrac1{x^2+1}$$

Now $\dfrac{2i}{x^2+1}=\dfrac1{x-i}-\dfrac1{x+i}$

Using this, $$\dfrac{d^n(1/x\pm i)}{dx^n}=\dfrac{(-1)^{n-1}n!}{(x\pm i)^{n+1}}$$

Now write $1=r\sin A,x=r\cos A\implies r=\sqrt{x^2+1},\cot A=x$

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