Write |exp(2z+i)| in terms of x and y

[ver_mathstats]

Homework Statement

Write |exp(2z+i)| in terms of x and y

Relevant Equations

|exp(2z+i)|

|exp(2z+i)| = e^2x, where did the imaginary part go?

|exp(2z+i)| = exp(2(x+iy)+i) = exp(2x+2iy+i) = exp(2x+i(2y+1)), I understand this but I'm confused how the solution is e^2x.

Could someone help explain it to me, thank you.

 

[FactChecker]

A property of exponents: $e^{(2x+i(2y+1))} = e^{2x}e^{i(2y+1)}$.
The factor $e^{2x}$ is the regular real exponential function that you should be familiar with. Can you work it from there? If not, you should review the definition and properties of the complex exponential function.

 

[ver_mathstats]

A property of exponents: $e^{(2x+i(2y+1))} = e^{2x}e^{i(2y+1)}$.
The factor $e^{2x}$ is the regular real exponential function that you should be familiar with. Can you work it from there? If not, you should review the definition and properties of the complex exponential function.

Yes I already know that, I was just asking why the imaginary number disappeared from the question. And I know Euler's formula of e^iy=cosy+isiny. But now I realized my mistake, I forgot | ... |.

 

[FactChecker]

I forgot | ... |.

Oh! That makes a lot of difference. :-)

 

[nuuskur]

Please mention explicitly that $z=x+iy$. Makes it incredibly confusing to read, otherwise. Either way, $|\exp (i\alpha)| = 1$ for any $\alpha\in\mathbb R$. So
$$|\exp (2z+i)| = |\exp (2x + i(2y+1))| = |\exp (2x)\exp (i(2y+1))| = \exp (2x).$$
Use properties of the modulus, where applicable.

 

[Mike S.]

Make sure you understand that complex numbers can be plotted as Cartesian coordinates (x, y) or as polar coordinates (r, theta). The second form emphasizes that the complex number has some modulus - a distance, like r, which increases "exponentially" when it is exponentiated - and a phase - some angle, that rotates around and around when exponentiated. The phase plots to a unit circle = cos theta + i sin theta, so it always has a modulus of 1, no matter how you rotate it.

 

[WWGD]

There is also the issue that lays out the difference between the Complex Exponential and the Real one:
$exp^{x +iy}=Cosy+iSiny =Cos(y+ 2k\pi)+i Sin( y+2k\pi) ; k \in \mathbb Z$ . Periodicity also explains the problem ( lack of injectivity) preventing a global logz inverse so that $e^{logz} \neq log e^z$ globally; issue of branches, etc. Edit2: Further down the road, you will see how, amazingly, the underlying Analytical properties are affected by the Topological properies.

Edit: I wish someone had given me this "Full Tour" before I got started.