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
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Why is $|1+e^{i\phi}|=|2\cos(\phi/2)|$?

$$|1+e^{i\phi}|=|2\cos(\phi/2)|$$ Hey guys, just wondering why the above is true, I don't think I quite understand how argand diagrams work. Supposedly, using an argand diagram I should be able to figure that out, but I'm not seeing it. Ultimately…
mugetsu
  • 155
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If $\left| \frac{z-i}{z-1}\right| = \sqrt2$ and $|z| = \sqrt 5$ and $Im(z)<0$, find $Im(z)$, $Re(z)$

I'm having a lot of problem with the first term. If I try to rationalize the fraction, I get an expression too long to work with. Similarly, substituting from the second term gives horrible results. Breaking the modulus apart and squaring gives me…
John Doe
  • 1,080
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2 answers

Complex Number, Quaternions and Octonions

There are complex $\mathbb C$, quaternions $\mathbb H$ and octonions $\mathbb O$. Is there any higher dimensional generalization of them, such in the $\mathbb R^{16}$? Or why do we just study three kinds of numbers in Mathematics? Any advice is…
gaoxinge
  • 4,434
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2 answers

Complex Numbers (Geometric Representations)

What is the geometrical interpretation of this operation: Multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$ Attempt: multiplication by −i = rotate by −π/2
No Name
  • 295
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7 answers

Why is the angle of $i^2 = \pi$?

On the complex plane , the angle of $i = \pi / 2$ and the angle of $i^2 = \pi$ . I understand that by definition $i^2 = -1$ but do not understand how to arrive at angle $\pi$ from $\pi / 2$ when we square $i$ on the complex plane. Can anybody…
NiallJG
  • 123
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3 answers

Determine a complex conjugate to $u(x,y)=x^3y-xy^3$

I know $\frac{\partial^2 u}{\partial x^2}=6xy$ and $\frac{\partial ^2 u}{\partial y^2} =-6xy$ and adding these together I get 0 which tells me they are harmonic functions. To determine the harmonic conjugate, I know that $\frac{\partial v}{\partial…
Al jabra
  • 2,331
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7 answers

Solving $z+i\overline{z}=iz-\overline{z}$

I want to solve $z+i\overline{z}=iz-\overline{z}$ ($\overline{z}$ is the complex conjugate). I have solved it setting $z=a+bi$. But can one solve without writing it $z$ a certain form, factorization maybe? Thanks in advance
user30523
  • 1,681
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Showing the limit does not exist

I am trying to show $\lim_{z \to 0} f(z)$ does not exist where $f(z)=\frac{xy}{2x^2+3y^2} +ix^2$. I am to show the limit does not exist by taking the limit along the straight line $y=mx$ where m is a constant. My plan is to show that the limit…
Al jabra
  • 2,331
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Separate imaginary and real parts from complex expression

I learned about complex numbers after I was trying to create a fractal object. The main problem is that I have an equation with complex numbers and I have to separate their parts (real & imaginary) to calculate the next iteration. Some equations…
user241514
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3 answers

How do you solve $\cos \pi z =0$?

How do you solve $\cos \pi z =0$? I am unsure what to do with the $\pi$. I know how to solve $\cos z = 0$, but $\pi$ is throwing me off. Can someone help start me off with this question please?
Katie Shaw
  • 81
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1 answer

Trouble with an easy complex equation

For the following equation: $\frac{Aj}{100\sqrt{2}} -\frac{A}{100\sqrt{2}} +\frac{x}{200}-\frac{xj}{200} =0$ where $A$ is a constant and $j$ is the imaginary unit. I thought the solution would be found by isolating the real and imaginary parts…
dylan7
  • 1,226
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4 answers

For complex $z$, find all solutions to: $(z - 6 + i)^3 = -27$

For complex $z$, find all solutions to: $(z - 6 + i)^3 = -27$ I reasoned that for this to be true, $z - 6 + i$ must be $= -3$ $\therefore z - 3 + i = 0$ $z = 3 - i$ However, there are two more solutions provided for this problem and I am not sure…
stariz77
  • 1,701
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3 answers

Find all complex numbers that satisfies this equation

Find all complex numbers that satisfies this equation $(z - 6 + i)^3 = -27.$ I found one of them being $ z = 3 - i $
chosenman
  • 25
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2 answers

Analyze $x^A(1 - x)^B$ divided by $(1 + x^2)$ with remainder $ax + b$

Assume $A, B \in \mathbb{Z}^+$. If $x^A(1 - x)^B$ is divided by $(1 + x^2)$, the remainder is $ax + b$, show that $a = (\sqrt{2})^B \sin\frac{(2A - B)\pi}{4}$ and $b = (\sqrt{2})^B \cos\frac{(2A - B)\pi}{4}$ So, I guess I have something…
stariz77
  • 1,701
2
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1 answer

How to solve a Complex equation involving the conjugate: $(x+iy)^2-2(x-iy)+1=0$

I want to find a Complex value for $z$ that satisfy the equation: $$z^2-2z^*+1=0$$ But i have never seen the conjugate taking part of an equation. What i have tried is give $z$ some components $x+iy$ So i have this: $(x+iy)^2-2(x-iy)+1=0$ And it…
Victor F Salazar G
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