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

learn more… | top users | synonyms

1
vote
2answers
68 views

Limit of $i^{n!}-2^{-n}$

I ran into this problem in Palka's Book which said to compute the limit of $z_n=i^{n!}+2^{-n}$. My approach was to consider the real and imaginary limits separately. Clearly the limit of the real part ...
1
vote
2answers
93 views

How find this $a$ such $(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$

Question: find all the complex $a$,such for every complex $z_{1},z_{2}(|z_{1}|,|z_{2}|<1,z_{1}\neq z_{2})$,such $$(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$$ My idea: ...
1
vote
2answers
353 views

Solving complex equation for z?

How do you solve equations involving $z = a + bi$ and imaginary units? The one I am looking at right now: $$\frac{z-2}{z+1} = 3i$$ If you could help me with this one, I think I can do the rest by ...
1
vote
2answers
28 views

Finding conjugate of a complex number

I am stuck with a really silly question : What is the conjugate of $a\bar c-\bar ac$ ? I calculated it as $\bar ac-a\bar c$ but according to my lecture notes, its conjugate is $a\bar c-\bar ac$ ...
1
vote
2answers
84 views

If $p(z)$ is a monic polynomial then $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$

I need some help with this problem: If $p(z)$ is a monic polynomial of degree $n$ then there is a $b\in\mathbb{C}$ such that $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$ where $z_1,z_2,\dots,z_n$ are simple ...
1
vote
1answer
105 views

Polar form of the sum of complex numbers $\operatorname{cis} 75 + \operatorname{cis} 83 + \ldots+ \operatorname{cis} 147$

The number $\operatorname{cis} 75 + \operatorname{cis} 83 + \operatorname{cis} 91 +\dots+ \operatorname{cis} 147$ is expressed in the form $r\operatorname{cis}(\theta)$, where $0\leq \theta< 360$. ...
1
vote
1answer
45 views

Prove of complex numbers inequality

Is it true that for any two complex numbers, say $a, b$, the following inequality holds: $|a\bar{b}| \leq |a|^2 + |b|^2$ ? How can we prove this?
1
vote
1answer
38 views

How find the minimum of the $|w^3+z^3|$,if $|z+w|=1,|z^2+w^2|=14$

let complex $z,w$ such $$|z+w|=1,|z^2+w^2|=14$$ find the minimum of the value $$|w^3+z^3|$$ My idea: let $$z=a+bi,w=c+di\Longrightarrow z+w=(a+c)+(b+d)i,z^2+w^2=(a^2+b^2+c^2+d^2)+2(ab+cd)i$$ then we ...
1
vote
2answers
79 views

Find $c$ if $a,b, \; c$ satisfy $c = (a+bi)^3 - 107i$

Find $c$ if $a,b, \; c$ are positive integers which satisfy $c = (a+bi)^3 - 107i$ I can try expanding the cube, but that seems too direct. What other ways are there to go about this?
1
vote
4answers
69 views

Which way will produce the following integral?

Which way $\gamma$ will produce the following integral? $$\int\limits_{\gamma}\frac{3+i}{z^5 - z}dz = 0$$
1
vote
3answers
160 views

Complex number equality

To Prove: $\displaystyle (\cos\theta +i\sin\theta)^4(\sin\theta-i\cos\theta)=\cos 8\theta+i\sin 8\theta$ My Attempt: $\displaystyle (\cos4\theta +i\sin4\theta)(\sin\theta-i\cos\theta)=\sin5\theta+i\...
1
vote
4answers
434 views

Is it true that “there is no such thing as the square root of minus one”?

Is the statement "there is no such thing as the square root of minus one" a true statement? It seems to me that we need to be careful about the word "the" as it appears in the statement. If we see it ...
1
vote
3answers
78 views

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(\...
1
vote
1answer
121 views

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)\...
1
vote
2answers
85 views

Proving a simple equation with complex numbers

Fix $A \in ℂ$ and $B \in ℝ$ Let $z \in ℂ$. Show that the equation $|z^2| + Re(Az) + B = 0$ has solutions iff $|A^2| ≥ 4B$ I have no trouble proving the forward implication, but its the "only if" ...
1
vote
3answers
45 views

Complex number with real part as 0

it is kinda of awkward, but is Equation: 0+3i=0? Or it simply means that it is imaginary number?
1
vote
2answers
60 views

Find real domain of a function results in $x \geq i$

I have an equation of the form $$f(x) = \sqrt{x^3 + x}$$ for which one needs to define the maximal domain, and image and domain are part of $\mathbb{R}$ (real numbers). $$x^3 + x \geq 0 \implies ...
1
vote
3answers
1k views

Solving a complex number equation with both $z$ and its conjugate $\bar z$

Determine all possible values of $z\in\mathbb{C}$ that satisfy the equation $4z = \overline{z}^2$. Where $\overline{z}$ represents the complex conjugate. (Hint: There are $4$ solutions.) ...
1
vote
1answer
73 views

Neat way to prove $\sin(\alpha+\beta)$ using complex exponential

I am supposed to prove that $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ using complex exponentials: $$ \begin{align} \sin\theta&=-\frac{1}{2}i(e^{i\theta}-e^{-i\theta})\\ \cos\...
1
vote
3answers
200 views

Real part of a quotient

Is there some fast way to know the real part of a quotient? $$\Re\left(\frac{z_1}{z_2}\right)$$ $z_i\in \mathbb{C}$
1
vote
4answers
621 views

Show $1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac{1}{2}+\frac{\sin[(n+1/2)θ]}{2\sin(θ/2)}$ [duplicate]

Show $$1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac12+\frac{\sin\left(\left(n+\frac12\right)θ\right)}{2\sin\left(\frac\theta2\right)}$$ I want to use De Moivre's formula and $$1+z+z^2+\cdots+z^n=\frac{z^{n+...
1
vote
3answers
160 views

how to solve circle dividing equation (complex numbers)

I have a equation that should divide a circle in even parts. As I found its called circle-dividing equation. I'v found same information how to solve a equation which has a form like this: $$z^6 = 1$$...
1
vote
3answers
192 views

Calculate Laurent series for $1/ \sin(z)$

How can calculate Laurent series for $$f(z)=1/ \sin(z) $$ ?? I searched for it and found only the final result, is there a simple way to explain it ?
1
vote
2answers
263 views

$z^2=-3-3i$ solve for $z$

Use De Moivre's theorem to solve the equation $z^2=-3-3i$. (Give your answers in polar form) Can you please explain why there are two answers? I cannot seem to understand why. By the way, the ...
1
vote
1answer
249 views

Closure of Algebraic Field to Complex Conjugation

I have an algebraic field $\mathbb Q(\gamma)$ with $\gamma$ the complex root of $X^3+X^2+X-1$, i.e., $\gamma\approx-0.771+1.115\mathrm i$. I have two closely related questions: Is $\mathbb Q(\gamma)...
1
vote
3answers
327 views

Why is the modulus of a complex number $a^2+b^2$?

Why is the modulus not $\sqrt{a^2-b^2}$? Carrying out standard multiplication this would be the result-why is this not the case? I know viewing the complex plane you can easily define the sum as ...
1
vote
1answer
90 views

Complex number inequality, $|e^{z_1}-e^{z_2}| \leq |z_1 - z_2|$ if $Re(z_1),Re(z_2) \leq 0$

I'm trying to show the complex inequality $|e^{z_1}-e^{z_2}| \leq |z_1 - z_2|$ holds if $Re(z_1),Re(z_2) \leq 0$. It seems intuitively obvious but I haven't been able to find something that works. ...
1
vote
2answers
148 views

An inequality problem with complex numbers

Knowing that $p$ and $q$ are complex numbers, $|p| < 1$ and $|q|<1$ show that $|\frac{p - q}{1 - q\bar{p}}| < 1$. I've tried to write: $p=x + yi$ and $q=a+bi$ which led me to $x^2 + y^2 + a^...
1
vote
1answer
90 views

Definition of complex number

In many situations (problems as well as solutions) I encounter the complex number $i$ which many times is defined as $i^2=-1$ instead of $i=\sqrt{-1}$, since it is "more preferred". Does anyone know ...
1
vote
2answers
219 views

Determine the integral $\int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$ using residues.

Determine the integral $$ \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$ using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications. In order to do this. We ...
1
vote
1answer
413 views

How to calculate an imaginary number to high exponent?

How can I calculate something like $(i+1)^{33}$ or similar high exponent without the use of a calculator?
1
vote
3answers
2k views

Complex numbers - finding minimum value

For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$ , find the minimum value of $|z_1-z_2|$ Can we go like this : Let $z_1 = x +iy$ therefore $|z_1| = \sqrt{x_1^2+y_1^2}$ and ...
1
vote
3answers
99 views

Adding real and imaginary parts

When trying to add $x$ to $x^{*}$ is it allowed to say that it would be equal to $2|x|$ i.e. so that $$x+x^{*}=2|x| $$ If this isn't the case is there any way to add them or should they be left as $$x+...
1
vote
2answers
13k views

How to determine if a matrix is positive/negative definite, having complex eigenvalues?

I am trying to deal with an issue: I am trying to determine the nature of some points, that's why I need to check in Matlab if a matrix with complex elements is positive or negative definite. After ...
1
vote
2answers
976 views

Imaginary part of a fraction

What is the imaginary part of this function? $$\displaystyle\frac{-2\mu(x)-2\mu(iy)}{x^{2}+2xiy-y^{2}+b^{2}}$$ I need just the imaginary part for the equation of streamlines in a fluids question but ...
1
vote
3answers
246 views

Placing complex polynomial into Taylor series form

For the following complex polynomial: Write the following polynomials in Taylor form centered at $z=2$: $z^{10}$ How come this simplifies to just a Taylor series of a binomial? Detailed explanation ...
1
vote
2answers
1k views

Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$ (\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger $$ and why not $ \phi^{\dagger} \psi^\dagger $. I want to know the ...
1
vote
3answers
943 views

How complex exponential converges and “sum of exponents” rule holds

How is it the complex exponential converges for any value of $z$ in the complex plane? $$e^{z} = 1 + \frac{z}{1!} + \frac{z^2}{2!} \cdots\cdots$$ How is it the "sum of exponents" rule holds for ...
1
vote
2answers
249 views

An inequality for two complex numbers

I recently saw the following inequality for complex numbers: If $a,b\in\mathbb C$ and $|a + b|$ and $|a-b|$ are each less than or equal to 1, then $$|a| + |b^2|/2 \leq 1.$$ How can one prove this?
1
vote
3answers
850 views

Expressing $e^z$ where $z=a+bi$ in polar form.

I am reading a passage of text that states: "We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
1
vote
2answers
174 views

How do you show $x_n=n(e^{\frac{2\pi i}{n}}-1)$ converges or not in C with the usual norm?

I have taken the limit of $x_n$ and got $2i\pi$. Now I am stuck trying to show $|n(e^{\frac{2\pi i}{n}}-1)-2i\pi|=0$. I am thinking I should try to write this in the form of $|a+bi|$ but I can't ...
1
vote
4answers
292 views

solve complex equation

$x^8 = \frac{1+i}{\sqrt{3} - i} = \frac{\sqrt[8]{\frac{2}{\sqrt{2}}}(\cos \frac{\pi}{4} + i \sin{\frac{\pi}{4}})}{2 \cos \frac{\pi}{6} + i \sin \frac{3\pi}{2}}$ What's the way to solve this kind of ...
1
vote
3answers
139 views

I don't understand this proof about Gaussian integers

Theorem: If p is a Gaussian prime and $p|zw$ for some gaussian integer $z,w \in Z[i]$ then $p|z$ or $p|w$. Suppose $p \not| z$ and lets deduce $p | w$. Let $u$ be a greatest common divisor of $p, z$....
1
vote
3answers
462 views

A contradiction involving exponents

Where is the error in the following statement: $i^2=(i^2)^{\frac{4}{4}}=(i^4)^{\frac{2}{4}}=(1)^{\frac{1}{2}}=1$? I feel the error is in the first equality, because $(i^2)^{\frac{4}{4}}$ is in fact $(...
1
vote
1answer
109 views

Polynomial equations in 2 variables with symmetry

Suppose $P(x,y)$ is a polynomial with real coefficients. Is it true that any solution $(x_0,y_0)$ of the system $P(x,y)=P(y,x)=0$ has the property that $y_0 = \overline{x_0}$ (i.e. they are conjugate),...
1
vote
1answer
761 views

Finding the real and imaginary parts of $\frac {z}{(1-e^z)^2}$

Could anyone help me find the real and imaginary parts of this $$ \frac {z}{(1-e^{z})^{2}} $$ where $z$ is complex? I can brute force it out but I'm worried that I'm missing an easier way, as I will ...
1
vote
1answer
76 views

Complex roots on a parallel to a bisector

This is from a collection book of problems on complex variables (Volkovyskii, Lunts, Aramanovich). I don't know how to tackle it without involving heavy unpromising calculations: Prove that both ...
1
vote
2answers
305 views

Multiplication and Division of Complex Numbers

this is a question on my homework that I am just lost with. Some direction would be greatly appreciated. Write $z_1$ and $z_2$ in polar form, and then find the product $z_{1}z_{2}$ and the ...
1
vote
3answers
235 views

Simplify formula $e^{-i0.5t}+e^{i0.5t}$

I want to ask, how can I simplify this formula ? $ e^{-i0.5t}+e^{i0.5t} $ I know that it can be simplify to $\cos(0.5t)$, but I don't know how :/
1
vote
2answers
40 views

Complex series should sum to zero but it's a puzzle

If we have a finite sum defined as $$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$ (where $k$ is an integer and $N$ is divisible by $4$), then how can we show that this sum is equal to $...