# Tagged Questions

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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### Why is the complex plane shaped like it is?

It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someome ...
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### Is this simple looking complex expression valid always?

$$z^{a+ib} = z^a*z^{ib} \hspace{2mm} \forall z\in \mathbb{C}$$ In high school I was always taught to see the + in complex numbers as analogous to that is reals. However can it be proven to be ...
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### Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right)$$ where I taken ...
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### For complex $z$, find the roots $z^2 - 3z + (3 - i) = 0$

Find the roots of: $z^2 - 3z + (3 - i) = 0$ $(x + iy)^2 - 3(x + iy) + (3 - i) = 0$ $(x^2 - y^2 - 3x + 3) + i(2xy -3y - 1) = 0$ So, both the real and imaginary parts should = 0. This is where ...
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### Triangle inequality for complex numbers

I just start to learn about complex numbers and I want to prove the triangle inequality, which says that if $z$ and $w$ are complex numbers, then $\displaystyle |z + w| \le |z| + |w|.$ My ...
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### Need help finding conjugate

$$\overline{z-2+4i} = 2z+3+8i$$ I got this question on my online assignment. I got to a point where I couldn't get rid of the conjugate of z and I don't know how to expand or what to do with it. I ...
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### How to sketch the region on the complex plane? [duplicate]

I am going through a basic course on complex analysis. I have a problem in understanding the following. E $\subset\mathbb{C}$ is defined as $$E := \{z\in\mathbb{C}:\vert z+i \vert = 2\vert z\vert \}$$ ...
Let $S$ be the unit disk without $0$. Find all $f \in Auto(S)$ I got the following idea. By Riemann 0 is a removable singularity. Since for $g\in Auto(D)$ where $D$ is the unit disk. \$g(z)= e^{i{\...