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I have been tasked to find the image of a set $U=\{z \in \mathbb{C} \mid \frac{-\pi}{2} \lt \Re(z) \lt \frac{\pi}{2} \}$ under the function $f(z)=\sin(z)$ which I have been asked to do so answering a series of sub-questions that go as follows:

(a): What is the image of the line segment $L_1=(\frac{-\pi}{2}, \frac{\pi}{2})$ (the real axis) under $f$?

(b): What is the image of the Imaginary Axis $L_2=\{iy \mid y \in \mathbb{R} \}$ under $f$?

(c): What is the image of the vertical line $L_3=\{\frac{-\pi}{2}+ iy \mid y \in \mathbb{R}\}$ under $f$?

(d): What is the image of the vertical line $L_3=\{\frac{\pi}{2}+iy \mid y \in \mathbb{R} \}$ under $f$?

(e): Given your observations in the previous steps what do you guess the image of the set $U$ is under $f$?

I am done figuring out the image of the invidual parts from (a)-(d) but ultimately cannot combine the results to judge the image of the set $U$ under $f$.

For parts (a)-(d), I got the following results:

(a): $(-1, 1)$

(b): $\{\iota \sinh(y) \mid y \in \mathbb{R}\}$

(c): $\{-\cosh(y) \mid y \in \mathbb{R} \}$

(d): $\{\cosh(y) \mid y \in \mathbb{R}\}$

How do I combine all the information obtained above into finding the image of the set $U$ under $f$. I also replaced $z$ with $x+\iota y$ and substituted it in the definition formula of the complex sine function and the addition formula and obtained the equivalent of $\sin(x+\iota y)$ as $\sin(x)\cosh(y)+\iota \cos(x)\sinh(y)$. How can I restrict the $x$ i.e. the $\Re(z)$ to being in the set $U$ so that I can plug in that information in the obtained equivalent of the complex sine function to obtain something meaningful?

Thanks

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The point about (e) is that boundaries map to boundaries under a continuous function. So (a) and (b) give some interior points and (c) and (d) help you demarcate the boundary.

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  • $\begingroup$ In particular, what's happening here is that the image $f(U)$ is the entire complex plane, except for "most of" the real axis; e.g. $\Bbb C-\{z\in\Bbb C : |\Re(z)|\geq 1\}$. [The part of the real axis that is included is the part calculated in (a), and parts (c) and (d) calculate the parts excluded.] $\endgroup$ – aleph_two Dec 19 '18 at 4:27

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