# Tagged Questions

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### Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
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### Absolute value of complex exponential

Can something explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook is telling me.) For example: $$|e^{-2i}|=1, i=\sqrt {-1}$$
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### Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
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### How does the complex exponential function transform the unit circle?

I know you can write every complex number on the unit circle as $e^{i\theta} = \cos(\theta)+i\sin(\theta).$ But what does it look like when you raise $e$ to the values? You get ...
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### Finding power series

I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$. I know that $1 + a + a² = 0$. I have tried to differentiate the expression and give values ...
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### Express $-1+i$ in exponential form.

Express $-1+i$ in exponential form. My attempt so far Let $z=-1+i$ $$r=|z|=\sqrt2$$ $$\theta=\tan^{-1}(-1)=-\frac{\pi}{4}$$ Now, this is where I go wrong (I don't know why it's wrong!): So in ...
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### Complex exponential

I know that the equation $e^{z}=-1$ has no solution had if been $z$ is a real number. So does the equation also has no solution when $z$ is complex?
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### Real solutions of the equation $\,z + e^{-z} - x = 0$ for $\,x > 1$ and $\,\Re(z) ≥ 0$

I have the complex equation $z + e^{-z} - x = 0$, where $z$ is complex and $x$ real, and I need to show that it has only one real solution for $x > 1$ and $\Re(z) \ge 0$. I do not see how this is ...
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### How does $e^{i x}$ produce rotation around the imaginary unit circle?

Euler' formula states that: $e^{i x} = \cos(x) + i \sin(x)$ I can see from the MacLaurin Expansion that this is indeed true, however, I don't intuitively understand how raising $e^{i x}$ power ...
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### Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.

This is kind of a big picture question. I just counted up all the symbols used in normal mathematics and, give or take, there are probably around 150 of them, tops. And that's really stretching ...
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### Exponential function and bijection

I am required to prove the following: For any real number $k$, prove that the exponential function $e^z$ is a bijection ($z$ is a complex number) from the strip $a < im z \leq k+2pi$ to the ...
### Real, imaginary parts of $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$
For the function $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$ , show that; $\qquad\mathrm{Re}\,G(w)=\sqrt2\sin^2(w/2)\quad$ and $\quad\operatorname{Im}\,G(w)=-1/\sqrt2\sin w$. I really need help with ...
The conjugate of $e^{-iwt}$ is $e^{iwt}$. Then, what would be the conjugate of $e^{iwt}$? Would it be $e^{-iwt}$? Also, for $|e^{iwt}|^2$, what would the value look like?