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.

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Operations with complex numbers to give real numbers

If: $|z|=|w|=1$ $1 + zw \neq 0$ Then $\dfrac{z+w}{1+zw}$ is real. How can prove that.
Jorge
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Cauchy-Riemann $w = |z^2|$

So for these types of questions, I can compute the partial differentials for Cauchy-Riemann but then I have trouble seeing/explaining where the function is differentiable? For example with this question I end up with $\partial u/\partial x =…
MathHelp1
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How to prove that :$\prod_{k=0}^{n-1}e^{\frac{2\pi i k}{n}}=(-1)^{n-1}\;\;\; n\in\mathbb{N}^*$

Can someone tell me how to prove the folowing equalty : $$\prod_{k=0}^{n-1}e^{\frac{2\pi i k}{n}}=(-1)^{n-1}\;\;\; n\in\mathbb{N}^*.$$ Thanks in advance.
user122327
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$z=(-1+i)^{11}+(-1-i)^{15}=?$

Can someone help me in this question : Let $z=(-1+i)^{11}+(-1-i)^{15}$ so $z=-96+160i$ $z=96-160 i$ $z=160-96i$ $z=-160+96i$ what is the right answer ? Thanks in advance.
user122327
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How to separate out real and imaginary terms

I have an equation like this: $$a+ib = \log(x+iy).$$ I need to separate the real and imaginary part in RHS so that I can equate the real part of LHS to real part of RHS and imaginary to imaginary part of RHS.
Shivji
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Can there be a complex line?

In an early math class, I was shown how all Reals could be constructed from Rationals using a 2-D representation (ex. Real numbers are represented by (a + b \sqrt{2} ) where a & b are Rational). While using the 'lesser' system of Rationals requires…
DGPIS
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Can some one explain to me what is going on here - power of complex number

So here is the question and the work to solve it, but I have no idea how one knows to do the first step or what the first step is... $$ \begin{align} (6-i\sqrt{12})^{12} &= \left[\sqrt{48}\left(\cos\left(\frac{\pi}{6}\right) -…
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Finding three roots of a complex number if we already know one root

If we know that $a+bi$ is one of the forth roots of the complex number $z$, how can we find the other three roots?
user155910
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Find the real and imaginary parts of $\sin(\frac{\pi}{2}+i\ln2)$

Find the real and imaginary parts of $$\sin\left(\frac{\pi}{2}+i\ln2\right)$$ Using the double angle formula I have gotten $$\sin\left(\frac{\pi}{2}\right)\cos(i\ln2)+\cos\left(\frac{\pi}{2}\right)\sin(i\ln2)$$ Which is then $$\cos(i\ln2)$$ But I do…
lar49
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Solve $(\frac{z+1}{z})^5 =1$ using fifth roots of unity

$$(\frac{z+1}{z})^5=1$$ Show that its roots are $$-\frac{1}{2}(1+i\cot(\frac{kπ}{5})), k = 1,2,3,4$$ I need to use the five fifth roots of unit, with angles $0,\frac{π}{5}, \frac{2π}{5},\frac{3π}{5},\frac{4π}{5}$ I started by doing…
George
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$\text{Im}(z)$ in equation

I'm having trouble with this equation: $$\text{Im}(-z+i) = (z+i)^2$$ After a bit of algebra i've gotten: $$1-\text{Im}(z) = z^2 + 2iz - 1$$ But i have no clue where to go from here, how do i get rid of the "$\text{Im}$"?
nooblet
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Principal root of 3+4i

Is there a neat way of writing the principal root of 3+4i? I have an answer, but it is very ugly. Thanks for any help in advance.
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Complex Numbers Exercise

If $a,b,c$ are complex numbers with $a+b+c=0$ and $\|a\|=\|b\|=\|c\| = r>0$ then prove that $$a^{2^n} + b^{2^n} + c^{2^n} = 0$$ Any ideas? Thanks!
bolzano
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locus of complex $z=(λ+3) + i\sqrt{3-λ^2}$

if $z=(λ+3) + i\sqrt{3-λ^2}$, for all real $λ$, then the locus of $z$ is ? Please help. Options are (A) circle (B) parabola (C) line (D) none of these
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What does $\theta = \text{arg}(a,b)$ mean?

I have this equation where an angle is calculated using following formula: $$\theta = \text{arg}(C_1, C_2)$$ where $C_1, C_2$ are some numerical values. What exactly does it mean?