Mathematics Stack Exchange
Mathematics Stack Exchange

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
1
vote
1 answer

Image of a line under a complex function: $f(z)=\frac{1}{\bar{z}}$

For a non-zero complex number $z$, let $f(z)=\frac{1}{\bar{z}}$. Let $w=f(z)$. As $z$ varies along the line $(1+2i)z-(1-2i)\bar{z}=i$ what curve does $w$ trace? I have tried by finding out the value of z in terms of $\bar{z}$ and then put it in the…
Sayantan
  • 187
1
vote
0 answers

How can I do this complex number question?

I've been struggling with this complex numbers question for some time now. I've tried converting each of the parts to cartesian form but it keeps cancelling out. Could anyone help please. Thanks in advance. Edit: I have found the line representing…
MathIsFun
  • 31
1
vote
1 answer

How does Collinearity Justify This?

I cannot make sense of one line in the given solution I am reading to this question: Problem: Let $A_0,A_1,\cdots,A_6$ be a regular $7$-gon. Prove that $\displaystyle \frac1{A_0A_1}=\frac1{A_0A_2}+\frac1{A_0A_3}$. Solution: Let $\varepsilon =…
user299140
  • 25
1
vote
0 answers

Name of theorem about replaceability of $i$ with $-i$

I recall, many years ago, reading that an true equation remains true if at all instances within the equation, one changes $i$ to $-i$ and $-i$ to $i$, however I cannot remember what this is called, or if it even has a name. If anyone is aware, I…
FizzKicks
  • 211
1
vote
1 answer

Why is the following solution valid to the given complex system of equations?

The question given is - what is the value of complex number z if it satisfies the following system of equations [w represents any complex number]: $z^3 + \overline{w}^{7} = 0$ $z^5w^{11} = 1$ Here, I understand that if one complex number = another…
Svee
  • 102
1
vote
1 answer

Proving complex polynomial coefficients are real/imaginary

I have two questions regarding complex numbers: Assume $\mathbb{\lambda, \overline\lambda}$ are solutions to the equation $\mathbb{z^2+az+b=0}$ and $\mathbb{\lambda}$ isn't real. are $\mathbb{a,b}$ real? Assume $\mathbb{\lambda,…
user3731180
  • 117
  • 4
1
vote
1 answer

A complex number and it's conjugate are a solution to an equation

I've been trying to solve this question for a few hours but I'm stumped, any guidance would be appreciated If $\gamma$ and $\bar{\gamma}$ are solutions to $z^2+az+b=0$ And $\gamma$ is not real then a,b are real Another question of the same vein If…
David Limkys
  • 135
1
vote
1 answer

Trouble with complex numbers

Is my following calculation true? $e^{a+ib}e^{\overline{a+ib}}=e^{a+ib}e^{a-ib}=e^{2a}$? for a,b real numbers or in general, what is $\overline{{z}^{w}}$ if $z,w$ are complex numbers?
user66906
1
vote
1 answer

Phase of a complex number

How can I calculate the phase of the following complex number? $ \omega, R ,C $ are some positive constants. $$ \frac{1-i\omega RC}{2+2i\omega RC} $$ In my book the answer is $ 2\arctan{(\omega\cdot RC}) $ But I cannot see why. Here is my…
FreeZe
  • 3,735
1
vote
1 answer

Complex equation with arg(z), but unknown on the form a+bi

$z=(x+i)^2$, $x > 0$ and a real number, solve for $x$ $\arg(z)=\frac {\pi}3$ $w = \sqrt z$ $w = x+i = \sqrt{x^2+1}$ $\ e^\frac{i\pi}{6}$ Now I could solve the bottom equation, but there must be an easier way? How should I have rather solved this…
Bob the Turtle
  • 127
1
vote
2 answers

How to solve the following question from complex number.

If $\alpha=e^{\frac{i2\pi}{7}}$ and $f(x)= a_0+\sum_{k=1}^{20} a_kx^k$ then the value of $$f(x)+f(\alpha x)+f(\alpha^2 x)+\cdots+f(\alpha^6 x)$$ is $ka_0$. Then find the value of $k$. I used a common property of complex numbers which is giving me…
sameed hussain
  • 386
1
vote
0 answers

Product rule for square roots of complex numbers

For real numbers, $\sqrt{a\cdot b} = \sqrt{a}\cdot\sqrt{b}$ only if at least one of $a$ or $b$ is greater than $0$. What does the corresponding rule look like for complex numbers? I understand that complex numbers aren't really comparable to each…
patentfox
  • 111
1
vote
4 answers

Square roots of complex numbers

I know that the square root of a number x, expressed as $\displaystyle\sqrt{x}$, is the number y such that $y^2$ equals x. But is there any simple way to calculate this with complex numbers? How?
Anonymous Pi
  • 1,297
1
vote
2 answers

Show that if $|z|=1$ then $|e^z|>\frac{1}{3}$

i knew that from modulus in complex, $|e^z|=|e^{x+iy}|\\=|e^x.e^{iy}|\\=|e^x|.|cos(y)+isin(y)|\\=|e^x|.\sqrt{cos^2(y)+sin^2(y)}\\=|e^x|$ and from $$|z|=1\\\sqrt{x^2+y^2}=1\\x^2+y^2=1$$ i don't know what to do from there, how can i prove it?
Ceciliads
  • 29
1
vote
2 answers

Finding all the solutions to a complex equation

I am asked to find all the solutions to $z^{42}=-1$. I go a head and square root both sides to produce $z^{21}= i$. Then I can write $z^{21}= r^{21} (\cos(21\ θ) + i\sin(21\ θ)) = 0+i. $ Hence $\cos(21\ θ)= 0$ and $\sin(21\ θ)=1$, as $r^{21} $ can't…
xiA
  • 436
1 2 3
…
99 100