Prove that for every $w \in \mathbb{C}$ it is true that $z + \sin z = w$ has exactly one root $z \in \mathbb{C}$.
The idea I have is to use Rouche's theorem.
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This is not true. It says that the function $f(z)=z+\sin(z)$ is a bijection from $\mathbb C$ to itself; in fact it's fairly well known and not hard to prove that the only holomorphic bijections of the plane are of the form $az+b$.
(Say $g\in H(\mathbb C)$. If $g$ is not a polynomial then it has an essential singularity at $\infty$, so Weierstrass tells us it cannot be injective, since the image of every neighborhood of $\infty$ is open and dense. Otoh if $g$ is a polynomial it cannot be injective if the degree is greater than $1$. So in fact the only injective entire functions are of the form $az+b$.)
Or in yet more detail: Say $g$ has an essential singularity, say at $0$. Say $0<r<|p|$, and let $V=g(D(p,r))$. Then $V$ is a nonempty open set. Let $W=g(\{z:0<|z|<|p|-r\})$. Then Weierstrass says $W$ is dense. Hence $W\cap V\ne\emptyset$, so $g$ is not injective.
David C. Ullrich
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1Why dont you show us the prove since it is not hard and fairly easy? – Guy Fsone Dec 12 '17 at 16:33
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I just did..... – David C. Ullrich Dec 12 '17 at 16:36
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You think Picards theorem is a random theorem? that is highly difficult if not the most difficult theorem I have ever met – Guy Fsone Dec 12 '17 at 16:37
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I cited Picard because it's well known. – David C. Ullrich Dec 12 '17 at 16:39
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By the way your answer seems correct I did not down vote you – Guy Fsone Dec 12 '17 at 16:40
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Fine. I just gave a more elementary sliighly longer argument. – David C. Ullrich Dec 12 '17 at 16:42
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Absolutely correct. – Mark McClure Dec 12 '17 at 16:46
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@GuyFsone Yes, the result of $W$ that I cite is elementary. Say $g$ has an isolated singularity at $p$. If the image of some deleted neighborhood of $p$ iis not dense then there exists $\alpha$ such that $1/(g-\alpha)$ is bounded near $p$. So $1/(g-\alpha)$ has a removable singularity, hence $g$ has a removable singularity or a pole. – David C. Ullrich Dec 12 '17 at 16:50
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There is a discussion of automorphism of $\mathbb{C}$ on this site. – swoopin_swallow Dec 12 '17 at 17:07
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People who downvote should comment explaining what the problem is – David C. Ullrich Dec 12 '17 at 23:05