Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

learn more… | top users | synonyms

2
votes
2answers
37 views

Argument of complex number $(\tan \theta)$

I'm given $-2+2\sqrt{3}i$. The question asks me to find the argument. My attempt, $\tan \theta=\frac{2\sqrt{3}}{2}$ So $\theta=\frac{\pi}{3}$. But the given answer is $\frac{2\pi}{3}$. Why?
1
vote
3answers
29 views

Understanding quotients of complex numbers

I am reading an old complex variables textbook which states: Given $z = a + bi$, $z_1 = a_1 + b_1i$, and $z_2 = a_z + b_2i \neq 0$, we have $z = \dfrac{z_1}{z_2} = \dfrac{a_1a_2 + b_1b_2}{a_2^2 + ...
3
votes
7answers
88 views

Show that $\cos(6x)= 32\cos^6x -48\cos^4x +18\cos^2x -1$

After writing down $\cos6x$= $Re (\cos x + i\sin x)^6$, I used the binomial theorem to expand the expression. Very soon it got really tedious and after trying $5$ times, fruitlessly, to arrive at the ...
0
votes
1answer
17 views

Derivative of a complex conjugate

I anticipate that this is a stupid question, but suppose $c \in C$. What is $\frac{\partial c^{*}}{\partial c}$? I've been trying and failing for about an hour to figure it out from the definition of ...
3
votes
1answer
51 views

Mandelbrot set of $c \cdot \cos(z)$

I'm given a task to write a program, that determines if a given point $c \in \mathbb{C}$ is in the Mandelbrot set of the function $$f_c(z) = c \cdot \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : ...
1
vote
2answers
32 views

Inequality $\left|z+w\right|\geq\left||z|-|w|\right|$ when $|w|\leq A|z|$.

Let $z,w\in\mathbb C$ and $|w|\leq A|z|$ for $A>0$. I want estimate from below $|z+w|$. I proceeded as follows. Since $$\left|z+w\right|\geq\left||z|-|w|\right|$$ and $-|w|\geq -A|z|$, I write ...
3
votes
1answer
15 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
1
vote
1answer
14 views

Contour integral of non analytic exponential function

The value of the integration of the function $f(z)$ over the circle of radius 3 centered at $z=1$, where $f(z) = e^{\frac{-1}{(z-1)^2}}$ this function has a pole at $z=1$ of $2$nd order. I don't ...
2
votes
2answers
25 views

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
2
votes
7answers
126 views

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 ...
3
votes
3answers
39 views

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 ...
3
votes
2answers
105 views

Guessing the other root to a quadratic equation

I just attempted to do the question below, but it seems that even after seeing the answer I'm not sure I understand the motivation for the solution. Let $\alpha ...
1
vote
1answer
42 views

Changing argument into complex in the integral of Bessel multiplied by cosine

I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind. I found the integral ...
-1
votes
0answers
22 views

Sketching the image of a circle under a complex polynomial

I want to sketch $w = z^3 + z^2 - iz + 1$ for $|z| = 2$. Finding the relation between $U(x,y)$ and $V(x,y)$ is my main question. I found $V^2 = (U - (x^3 + x^2 + 1))^3$ but I don't know how to use ...
1
vote
7answers
77 views

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 ...
0
votes
3answers
40 views

Solution of an equation with complex numbers [on hold]

Knowing that $2+i$ is a solution of $z^3 - 11z + 20 = 0 $ Calculate the other solutions
0
votes
3answers
68 views

prove that $f(z)+f(iz)=0$ please

When $f(z).f(iz)=z^2\space \forall z \in \mathbb{C}$ How to prove that $f(z)+f(iz)=0 \ \ \space \forall z \in \mathbb{C}$ I try Let $f(z)+f(iz)=M$ $ f(z)=\frac {z^2}{f(iz)}$ ...
-2
votes
1answer
16 views

Find the Cartesian equation of the Locus [on hold]

$$f(x)= \frac{3x+4}{x-3} =x^*$$ where $x^*$ is the conjugate of $x$ Find the Cartesian equation of the Locus that satisfy $$f(x)=x^*$$
0
votes
2answers
31 views

Problem about complex number

Find all values of $(-1)^{1/3}$ I used the identity's and such and got a part where I got $e^{1/3\log(-1)}$, and I'm not sure how to do the next step and get to the answer. Can anyone send in the ...
1
vote
1answer
21 views

Find the values of $a,b,c$ of the complex function $f(x)= (ax+b)/(x+c)$

The task is to find the values of $a$,$b$, and $c$ of the complex function $f(x)=\frac{ax+b}{x+c}$ where $a,b,c \in \mathbb{R}$. It is given that $f(2i)=-2i$ and $f(1+3i)=1-3i$. I tried to make an ...
1
vote
1answer
18 views

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 ...
4
votes
3answers
43 views

Express $w=f(z)=\frac{1}{(1-z)^2}$ in the form $w=u(x,y)+iv(x,y)$

I start by writing $f(z)$ as $$\frac{1}{(1-(x+iy))^2}$$ and then I expand the bottom to get $$\frac{1}{(1-2x+x^2-y^2) + i(2y-2xy)}$$ The answer says ...
5
votes
1answer
77 views

Intuition behind $i^{i}$.

My query is about the $i^{i}$ , where $i$ is defined to be the imaginary unit, and $i \in C$. I know the proof of this value, we just have to substitute $i$ as ...
1
vote
0answers
28 views

Finding the RHS of a complex equation

Consder $z$, a complex number, such that $z+\frac{1}{z} = 2\sin(a)$, $a \in (0,2 \pi)$ . Find : $$ z^{4n} + \frac{1}{z^{4n}} = ? $$ I tried expanding it in trig form, then by applying de Moivre's ...
2
votes
1answer
24 views

How does uncertainty propagate through an equation with complex variables?

I am trying to understand how uncertainty propagates through systems with complex variables. Given the general error propagation formula $$ \sigma^2_u = \left(\frac{\partial u}{\partial ...
7
votes
2answers
88 views

How to visualize $\mathbb{C}^2$?

In a homework question I had to do, the rotational matrix $A = \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}$ was given. Its eigenvalues in $\mathbb{C}$ are $i$ and $-i$. The set of all eigenvectors ...
-4
votes
1answer
21 views

Find the priniciple argument of $-5\sqrt{2}+5i\sqrt{2}$ [on hold]

Find the priniciple argument of $-5\sqrt{2}+5i\sqrt{2}$ must be the exact value and in the form $z=...\pi$
0
votes
0answers
9 views

Is there a particular order when evaluating the complex exponential?

This question might seem silly. But I just cannot get this: $e^{i 6\pi} = 1+0i$ ($e^{i 6\pi})^{1/2} = \sqrt{1+0i} = \sqrt{1} = 1$ ($e^{i 6\pi})^{1/2} = e^{i3\pi} = -1+0i = -1$
-1
votes
1answer
38 views

If $\sin{x}+\sin{y}+\sin{z}= \cos{x}+\cos{y}+\cos{z}=0$, find the value of $\cos{2x}+\cos{2y}+\cos{2z}$. [on hold]

Is there any way to solve this question using complex numbers? I am trying the general way too but I am unable to solve the question.
1
vote
5answers
47 views

Complex Numbers Roots of Unity

By multiplying two roots, one is the conjugate of the other, we get one. Does someone know why and proof that. Many thanks
18
votes
5answers
1k views

Proving the following number is real

Let $z_i$ be complex numbers such that $|z_i| = 1$ . Prove that : $$ z\, :=\, \frac{z_1+z_2+z_3 +z_1z_2+z_2z_3+z_1z_3}{1+z_1z_2z_3} \in \mathbb{R} $$ This problem was featured on my son's final ...
0
votes
1answer
11 views

How to find the complex potential for the following flow under certain conditions?

We've used $z=i(Z+4/Z)$ as a conformal mapping to map the exterior of a circle $|Z|=2$ to the exterior of the line segment $(-4i,4i)$. We now want to write the complex potential of the uniform flow ...
0
votes
1answer
31 views

Hilbert's inequality for $\left|\sum_{n,m}a_n \bar a_m\right|$.

We know that, an Hilbert's inequality states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. Then write an inequality ...
7
votes
4answers
631 views

What is the motivation for complex conjugation?

I have been dealing with complex numbers for few years now. But when I've tried to think about the motivation behind complex conjugation, I was not sure. Let me write what I am working with. For a ...
1
vote
1answer
34 views

Find a Linear Fractional Transformations (LFT) $w(z)$

I have absulotly no idea how to approach this question, Can anyone please provide with a hint or any kinda information so I can solve this question. Thank you very much for you help
-2
votes
2answers
42 views

Cauchy-Riemann equations Complex Numbers [on hold]

I have used the theorem if f'(z) = 0 then f(z) is a constant. I have proved it by using Cauchy Riemann's theorem. b
3
votes
2answers
34 views

Complex Differentiation

Can anyone give a hint to how to approach this question?
4
votes
3answers
143 views

Find$\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 2x + 4}\,dx$ and $\int_{-\infty}^{\infty} \frac{\sin(x)}{x^2 + 2x + 4}\,dx$

Find $$\int_{-\infty}^{\infty} \dfrac{\cos(x)}{x^2 + 2x + 4}\,dx$$ and $$\int_{-\infty}^{\infty} \dfrac{\sin(x)}{x^2 + 2x + 4}\,dx$$ I find it really difficult. Much appreciate it if anyone can ...
-4
votes
1answer
32 views

What does $\textrm{Im}(z) + 1$ mean?

Complex Numbers What does $\textrm{Im}(z) + 1$ mean? Does it mean the imaginary part $+ 1$ ? To get $(z+i)$? Or does it mean something else such as $+1$ is the real part?
-3
votes
0answers
12 views

use De Moivre's theorem to show that: cos(3theta) = 4cos(3theta)-3cos(theta) and sin(3theta) = 3sin(theta) - 4sin(3theta)

let z = cos(theta) + isin(theta) a) find z^3 using binomial expansion b) use De Moivre's theorem to show that: cos(3theta) = 4cos(3theta)-3cos(theta) and sin(3theta) = 3sin(theta) - ...
0
votes
0answers
33 views

how can I show that $\cot\pi z$ and $\csc \pi z$ have simple poles for every integer $n$? so then I can calculate residues at those poles?

how can I show that $\cot\pi$z and $\csc\pi$z have simple poles for every integer $n$? so then I can calculate residues at those poles?
1
vote
1answer
28 views

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 & ...
0
votes
1answer
35 views

Find the maximum & minimum value of complex number.

Let $z_1, z_2, z_3, \ldots, z_{13}$ be real numbers, & let $A$ be the average of complex numbers $[e^{iz_1}, e^{iz_2}, \ldots ,e^{iz_{13}}]$, where $i=\sqrt{-1}$. As the value of z's vary over ...
3
votes
0answers
78 views

Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
2
votes
2answers
45 views

Are complex differentiable function in a point analytic?

I know that if a function $f$ is complex differentiable in a neighborhood of $z_0$, then we say it's holomorphic in $z_0$ and it's also analytic in a neighborhood of $z_0$. But suppose that I know ...
3
votes
4answers
45 views

Simplifying quartic complex function in terms of $\cos nx$

$$z= \cos(x)+i\sin(x)\\ 3z^4 -z^3+2z^2-z+3$$ How would you simplify this in terms of $\cos(nx)$?
1
vote
1answer
24 views

Taking the complex conjugate of some complicated composite function

I'm aware of the rule where to take the complex conjugate of anything, you simply replace any $i$'s with $-i$, and to conjugate any composed functions (i.e. $f*(g(z))=f(g*(z)))$ What is the ...
0
votes
2answers
31 views

algebra ring morphism [closed]

I have this problem which says Let there be $f\colon \mathbb{Z}[i]\to \mathbb{Z}_5$; $f(a+bi) = \widehat{a+3b}$. a) How do I show that $f$ is a ring morphism? b) Determine $\operatorname{Ker}f$ ...
0
votes
0answers
20 views

Where does $\cos z$ conjugate is holomorphic

I solved this question using Cauchy-Riemann equations and got a contradiction,meaning not analytic everywhere, but I am not sure I am right. Got from 1 equation $x=\pi k$ from the other $x=\pi/2+\pi ...
1
vote
0answers
25 views

How find set $f(\Bbb R^3)$ for $f(x, y ,z)= e^{i(x+y+z)} + e^{i(x-y-z)}+ e^{i(-x+ y-z)}+ e^{i(-x-y+ z)}$?

Let $f:\Bbb R^3 \to \Bbb C$ such that $f(x, y ,z)= e^{i(x+y+z)} + e^{i(x-y-z)}+ e^{i(-x+ y-z)}+ e^{i(-x-y+ z)}$. How can one find the set $f(\Bbb R^3)$?