Questions tagged [complex-numbers]

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

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
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Derivative complex function

Calculate the derivative of $f(z)=\frac{(1+z^2)^4}{z^2}$ I know that this function is discontinous at $z=0$, what I did is just calculate the derivative the same manner as is done with real functions. $$\begin{align}f'(z)…
Roland
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Show that $\frac{1+\cos 5x+i\sin 5x}{1+\cos 5x-i\sin 5x}=\cos 5x+i\sin 5x$

$$\frac{1+\cos 5x+i\sin 5x}{1+\cos 5x-i\sin 5x}=\cos 5x+i\sin 5x$$ When I attempted this I first tried multiplying top and bottom of the LHS by the complex conjugate of what's on the bottom, $1+\cos 5x+i\sin 5x$. After simplification I…
RobChem
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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 I got stuck since there are two unknowns for each…
stariz77
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I am getting $i^{-1}=\pm i$.

I am trying to find $i^{-1}$. I already know that the answer is $-i$, but I can't figure out a way to determine that using math. This is what I am doing: $$i^{-1}$$ $$\frac1i$$ $$\sqrt{\left(\frac1i\right)^{2}}$$ $$\sqrt{\frac{1}{-1}}$$…
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Proof regarding complex numbers - how to continue?

Let $z,w\in\mathbb{C}$ and $|z|,|w|<1$. Show that $\displaystyle \left|\frac{z-w}{1-z\bar{w}}\right|<1$. My…
Galc127
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Using $\arcsin(z) = -i\log(i(z + \sqrt{z^2 - 1}))$ to compute $\frac{d}{dz}\arcsin(z)$

I have to use $\sin^{-1}(z) = -i\log(i(z + \sqrt{z^2 - 1}))$ to compute the derivative of $\sin^{-1}(z)$, $\frac{d}{dz}\sin^{-1}(z)$. Here is my process: $$\sin^{-1}(z) = -i\log(i(z + \sqrt{z^2 -1}))$$ $$\sin(-i\log(i(z + \sqrt{z^2 - 1}))) =…
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How does one find $z,w,\lambda \in \mathbb{C}$ such that $(zw)^{\lambda}\neq z^{\lambda}w^{\lambda}$?

I want to find $z,w,\lambda \in \mathbb{C}$ such that $(zw)^{\lambda}\neq z^{\lambda}w^{\lambda}$. I wasn't able to find an example so far. If you take $z$ and $z^{-1}$, then $(zw)^{\lambda}=z^{\lambda}w^{\lambda}$. If $z=1+i, w=-1+1$ and…
user23505
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Find the value of $z^n+1/z^n$ if $n$ is arbitrary natural number

If $z$ is a complex number satisfying $$z + \frac{1}{z} = \sqrt{3}$$ then for arbitrary natural number $n$, determine the value of $$z^n + \frac{1}{z^n}$$ I have tried it with $n=2,3,4$ but it came to different answers.
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How does one geometrically describe the set $S=\{z\in\mathbb{C}:\;|\operatorname{Arg}(z-i)|\lt\frac{\pi}{6}\}?$

This is Exercise EP$10$ (g) from Fernadez and Bernardes's book Introdução às Funções de uma Variável Complexa (in Portuguese) How does one geometrically describes the set $S=\{z\in\mathbb{C}:|\operatorname{Arg}(z-i)|\lt\frac{\pi}{6}\}?$ I tried to…
user23505
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Principal argument of $-2i$

How to write the principal argument of $-2i$ ? I cannot just write $-\pi/2$, altough it is obvious, I have to justify it somehow. Can I say that $\lim\limits_{x\to 0}\arctan\left({\frac {-2}x}\right)$ ?
OBDA
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Find the locus of $w$

$$ \text{Find the locus of $w$, where $z$ is restricted as indicated:} \\ w = z - \frac{1}{z} \\ \text{if } |z| = 2 $$ I have tried solving this by multiplying both sides by $z$, and then using the quadtratic equation. I get $z = \frac{w \pm…
Sam Chahine
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how does this expression cancel out

How does $$\frac{2t-1-i}{2t^2-2t+1}=\frac{1+i}{-1+(1+i)t}$$ I just can't see how this works... I typed the LHS in WFA and it gave the RHS but I don't know how anything can cancel.
snowman
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Find all complex numbers satisfying the equation

Find all complex numbers $z$ satisfying the equation $$\left|z+\frac{1}{z}\right|=2.$$
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How come complex numbers encompass all the numbers we need?

Are there numbers other than complex numbers? for example, \begin{eqnarray} |x| = -1 \end{eqnarray} Surely, the equation does not make much sense initially since by definition magnitude is positive. But in the previous times when people knew only…
chatur
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