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I have tried searching the internet for any possible solutions to try and find an answer to this question. One idea that I had was to use Hyperbolic functions to try and get the form $z = u + iy$. But the question is:

$$e^{\sin(\sin (z))}$$

I couldn't get past the 'e' part. Not only that, I am having trouble with the hyperbolic substitutions and simplifying them in a way that results in $z = u + iy$. Is there another way to show whether this expression is analytic or not? Or is it possible to prove it through $U_x=V_y$ and $U_y= - V_x$ ?

Do forgive me if it my question is a bit confusing.

Hanul Jeon
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haziq aiman
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    Do you know that $\exp$ and $\sin$ are analytic in $\Bbb C$? Do you know that compositions of analytic functions are analytic again? – Martin R Apr 19 '20 at 08:31
  • Switching to real and imaginary parts is very unlikely to produce an algebraic expression that strikes you immediately as "holomorphic", because of the general fact that it never does: if $f$ is a nice function, then $f(\Re z)$ is basically never holomorphic. If you want to use Cauchy-Riemann, then you can and seeing the function in terms of real and imaginary parts of $z$ is more or less needed. –  Apr 19 '20 at 08:36
  • I actually do not know that. I am relatively new in this topic, so I am a bit confused on the relationships of analytic functions. Thank you for the information. (Martin R) – haziq aiman Apr 19 '20 at 08:38
  • Can you please explain what 'holomorphic' means? (Gae. S.) – haziq aiman Apr 19 '20 at 08:41
  • you can get specific people's attention by typing @ and then their username. Like @haziqaiman – Calvin Khor Apr 19 '20 at 08:51
  • @haziqaiman Holomorphic (on some domain) means that the complex derivative exists (on said domain). But for the purpose of this post you could think as if I said "(complex-)analytic". –  Apr 19 '20 at 09:14
  • @CalvinKhor Thank you so much. – haziq aiman Apr 20 '20 at 02:11
  • @Gae.S. I see. I think I understand this better now. Thank you. – haziq aiman Apr 20 '20 at 02:12

2 Answers2

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Both $e^z$ and $\sin z$ are entire (i.e. analytic everywhere) on $\Bbb C$, so is any order of their combination.

Mostafa Ayaz
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Hint: You can use this version of Cauchy-Riemann instead:

$$\frac{\partial f}{\partial x} = -i\frac{\partial f}{\partial y}$$

Ninad Munshi
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