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For what $z\arcsin(z)$ will be a real number? I tried to discover function $\sin(z)$ and find all $$z = a + b\cdot i: \sin(z) = Re \Rightarrow \arcsin(z) = Re.$$ But, of course, it's not correctly. From wolframalpha I found that for $$z>1 \quad \arcsin(z) = a + b\cdot i.$$ How to prove it? Are there any other restrictions for $z$? For example, must z be only real number?

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up vote 5 down vote accepted

Let $\sin ^{-1}(z)=a$, where a is a real number, $a\in \mathbb{R}$.

Then $$\sin(\sin ^{-1}(z))=\sin (a)=z$$ again where a is real. So $z\in [-1,1]$.

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