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Complex analysis,finding image under mapping

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Find the image of the strip R = { z in complex with 0 < Re(z) <$$\pi$$ } under f(z)=sin(z).

Oct 25, 2018

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It looks like it maps to the strip

$$\{z: 0 \leq Re[z] \leq 1,~Im[z] \in \mathbb{R}\}$$

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Oct 26, 2018
#2
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This is incorrect.

The definition of sin(z) for complex numbers is:

$$sin(z)=\frac{{e}^{iz}-{e}^{-iz}}{2i}$$

Guest Oct 26, 2018
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The mapping f(z)=sin(z)=sin(x+iy)=sin(x)cosh(y)+icos(x)cosh(y)  map z to Real(f(z)) in positive real number and Im(f(z)) to real, which map to first and fourth quaduant and real axis of complex plane. That is my simpilfie answer and this probelm was one of test question,and I am doing test correction. Does anyone have any thoughts on this?

Oct 26, 2018
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I dont understand, are you saying that the image of the strip R under sin(z) is the first quadrant+the fourth quadrant+the real axis?

This answer is wrong (If you know what the open mapping theorem is, you can see how the theorem shows that this can't be the image, and that the image has to be an open set)

Also keep in mind that sin(x+iy)=sin(x)cosh(y)+icos(x)sinh(y), not sin(x)cosh(y)+icos(x)cosh(y)

Guest Oct 26, 2018
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Thanks guest fort spoting my mistake. I am sorry, I was doing another homework and didn't double check my typing.Yeah , you are right，the correct formula for sin(z) is sin(z)=sin(x+iy)= sin(x)*cosh(y)+i*cos(x)sinh(y). The image of strip R under mapping f(z)=sin(z) is  quadrant+the fourth quadrant+the positive real axis of complex plane. Let w=f(z),then Re(w) is the set positive number and Im(w) is real number.Then the image is an open set,isn't it? Where the definition of open set is  A set is open if all its points are interior points,and the definition of interior point is A point a ∈ G is an interior point of G if some open disk with center a is a subset of G. And open disk is the inside of this circle with center a and radius r; we use the notation D[a,r] := {z ∈ C : |z − a| < r} .

See a similar problem posted on stack exchange

https://math.stackexchange.com/questions/1082250/determine-the-image-of-the-strip-s-consisting-of-all-points-z-with-frac

See my instuctor's answer to his sample test problem

fiora  Oct 26, 2018
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Yes I think your answer is correct  (The image is the right half plane)

I followed your link but question 5 is different, am I missing something?

Guest Oct 27, 2018
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Thanks for verification,guest. My professor like to give the student some wired question.He is differently something else and very infamous in my University. He even taught the co-chair of Department of Math & Computer Sicence of my University,so you can imagine his age. I am pretty sure that sample problem in the link is the best and the only sample I can found.Thanks again for your suggestion.

fiora  Oct 29, 2018