# Questions tagged [complex-numbers]

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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### For a matrix $A(z)$ that represents the operation of multiplication with a complex number $z$, what does it mean for $e^{A(z)t} = A(e^{zt})$?

We can have a complex number $z = a + bi$ that determines a matrix $A(z)$ in the following way: $$A(a + bi) = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$ This matrix represents the ...
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### Polar form of the sum of two complex numbers.

I was trying to express the addition of two complex numbers only in function of its absolute values and arguments while preserving the result in polar form, with the fewest terms possible, for ...
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### Explanation for why $(-1)^{-i} = e^\pi$?

I can't find any explanations online as to why the following holds: $$(-1)^{-i} = e^\pi$$ I assume there's a simple explanation pursuant to the rules of complex number manipulation, but is there any ...
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### Introduction to the Binary Tetrahedral group and the 24-cell

Context and introduction I was playing with complex number sequences $Z_n=r_n\omega^n=u_n+iv_n$ represented in space and realized that it's always possible to associate up to 48 naturally symmetric ...
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### How to find principal value of the cubic root?

I tried to find principal value for $\sqrt[3]{z}$ , I started from $$z=w^3$$ So $$w_1=\sqrt[3]{r} \exp\left(\frac{Arg(z)}{3} i\right)$$ $$w_2=\sqrt[3]{r} \exp\left(\frac{Arg(z)+2\pi}{3} i\right)$$ ...
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### Replacing z with its conjugate to solve questions in complex numbers [closed]

I have noticed that in many questions of Complex numbers , specially those involving a polynomial in z , we replace z with zbar and solve the question, what's the rationale behind doing this. And does ...
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### The function $f$ is given by the formula $f(z) = \frac{1 - \cos z}{z^5 + z^7}$

The function $f$ is given by the formula $$f(z) = \frac{1 - \cos z}{z^5 + z^7}.$$ (a) Classify the singularity at the point $z = 0$ and write down the principal part of the Laurent series ...
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### Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.

Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$. (a) Determine and classify the singular points of the function $f$ and calculate the residues at these ...
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### Minimum value of $|z^4+z+\frac{1}{2}|$ on the unit circle

Let $z$ be a complex number. What is the minimum value of the expression $|z^4+z+\frac{1}{2}|$ for $|z|=1$? I wanted to explore the long process of considering $z=x+iy$, and substituting to get the ...
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### What's the history of defining the amplitude of a complex number as its argument(phase)?

I see some confusing definitions saying the amplitude of a complex number is the argument or phase of the complex number, as in the following examples. What is the amplitude of a complex number? ...
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