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

0
votes
1answer
45 views

Sum of all elements that solve $z^n* = 1$

Let $n>1$ be a fixed positive integer and let $H=\{z \in\Bbb{C} |z^n=1\}$ Prove that the sum of all elements of $H$ is $0$. I've been struggling on this question for far too long for how simple ...
5
votes
1answer
64 views

Evaluate the integral $\int_0^{2\pi} \frac{d \theta}{5-3 \cos \theta}$

$$\int_0^{2\pi} \frac{d \theta}{5-3 \cos \theta}$$ My attempt: Let $z=e^{i\theta}$ which gives $d\theta = \frac{dz}{iz}$ Thus, $$\oint_C \frac{1}{5-3(\frac{z+z^{-1}}{2})}\frac{dz}{iz}$$ ...
2
votes
1answer
73 views

Evaluate the integral using the theory of residues: $\int_0^{2\pi} \frac{(\cos \theta)^2 d \theta}{3-\sin \theta}$

$$\int_0^{2\pi} \frac{(\cos \theta)^2 d \theta}{3-\sin \theta}$$ I''m having trouble simpliyfing this into a form that will allow me to use Residue Theorem. I got it to the point where the integrand ...
0
votes
1answer
39 views

Holomorphicity of $f(x + iy) = x^2 + iy^2$

By definition: $f: E \rightarrow \mathbb{C}$, where $E$ is an open subset of $\mathbb{C}$ is holomorphic on $E$ if $f$ is $\mathbb{C}$-differentiable at all points of $E$. The key point being ...
1
vote
2answers
56 views

Least value of $|z-w|$

On an argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing complex numbers $w$ satisfying arg$(w-2)=\frac{3}{4}\pi$ find the least value of ...
0
votes
0answers
12 views

The domain of the Digamma function and its extension

First, we know that $\Gamma(x)>0$, for all $x>0$, so define $\psi(x)=\frac{\mathrm{d} }{\mathrm{d} x}\ln(\Gamma(x))=\Gamma'(x)/\Gamma(x)$, for all $x>0$. This is the Digamma function. It is ...
0
votes
0answers
19 views

Notation for variables representing complex numbers

Is there a standard way to indicate that a variable represents a complex number? In physics, it is convenient to analyze oscillating systems using complex numbers. The authors of one popular ...
1
vote
3answers
33 views

Complex conjugates

If $z=e^{2 \pi i/5}$, so that $z=x+iy$ where $x=\cos(2\pi i/5)$ and $y=\sin(2\pi i/5)$, then how are $z$ and $z^4$ complex conjugates with each other? I see that visually, but $$z^4=x^4-6 x^2 ...
1
vote
2answers
52 views

Cauchy–Schwarz inequality in Complex variables

I have seen various proofs for Cauchy–Schwarz inequality but all of them discuss only of real numbers. Can someone please give the proof for it using complex numbers in simple steps?
4
votes
3answers
98 views

Find $a^{100}+b^{100}+ab$

$a$ and $b$ are the roots of the equation $x^2+x+1=0$. Then what is the value of $a^{100}+b^{100}+ab$? Here's what I found out: $$a+b=-1$$ $$ab=1$$ but how to use this to find that I don't know! ...
2
votes
2answers
140 views

Simplifying this (perhaps) real expression containing roots of unity

Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant. Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$. ...
3
votes
2answers
46 views

$z=e^{2\pi i/5}$ solves $1+z+z^2+z^3+z^4=0$ [duplicate]

What is the best way to verify that $$1+z+z^2+z^3+z^4=0$$ given $z=e^{2\pi i/5}$? I tried using Euler's formula before substituting this in, but the work got messy real fast.
0
votes
0answers
21 views

Find a f(x) that is reducible in a field $F$ that has no root in $F$

Find a $f(x)$ that is reducible in a field $F$ that has no root in $F$ I'm trying to find a polynomial are reducible in an arbitrary field $F$ but has no root in $F$
1
vote
0answers
20 views

Calculation of partial derivatives of real and imaginary parts of $g(z)=(rez \times imz)^{1/3}$

Let $g(z)=(rez\times imz)^{1/3}$. Calculate partial derivatives for real and imaginary parts, state where they exist and at which points they do not exist. Verify Cauchy Riemman equations at ...
3
votes
2answers
50 views

Showing convergence with complex numbers

I would like to show that if $|a-b|<\delta$ then $|e^{a}-e^b|<\epsilon$. Where $a$ is complex and $b$ is real. In essence if the difference between a and b is small then the difference between ...
1
vote
0answers
28 views

Limit of a complex number raised to a power

This question has been asked and answered here:limit when it exists of a complex number raised to an integral power but I don't understand why this is so and I can't comment on the original question. ...
22
votes
7answers
2k views

How to prove that a complex number is not a root of unity?

$\frac35+i\frac45$ is not a root of unity though its absolute value is $1$. Suppose I don't have a calculator to calculate out its argument then how do I prove it? Is there any approach from ...
0
votes
1answer
26 views

Suppose $Z_1$ and $Z_2$ are complex numbers such that $\frac{Z_1-2Z_2}{2-Z_1\bar Z_2}$ is unimodular. Where does $Z_1$ lie?

Question Suppose $Z_1$ and $Z_2$ are complex numbers such that $\frac{Z_1-2Z_2}{2-Z_1\bar Z_2}$ is unimodular. Where does $Z_1$ lie? My try given that $\left|\frac{Z_1-2Z_2}{2-Z_1\bar Z_2}\right|= ...
1
vote
1answer
25 views

If $Z$ is a complex number such that $ |Z|\ge 2 $ the the range of values of $\ |Z+\frac{1}2|\ $ is?

If Z is a complex number such that $ |Z|\ge 2 $ the the range of values of $\ |Z+1/2|\ $ is ? My try I tried using the triangle inequality according to which ||Z| +|1/2|| $\ge$ |Z+ 1/2| this ...
1
vote
2answers
33 views

$z^4 = w$ has one solution $z = x + yi$ how do I find the other 3 solutions

As the title states, $z^4 = w$ has one solution $z = x + yi$ how do I find the other 3 solutions $x,y$ are real numbers
1
vote
1answer
22 views

Use a rectangular contour to evaluate the integral

$$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$ I need to evaluate the above integral specifically using a rectangluar contour and at some point ...
1
vote
2answers
21 views

Solving inequality in complex plane

I have to graphically represent the following subset in the complex plane being z a complex number: $A={1<|z|<2}$ However after trying to do it on WolframAlpha it says that "inequalities are ...
0
votes
1answer
16 views

Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
0
votes
0answers
22 views

Find the order of the pole and the residue of $f(x)=\frac{\cos z}{z^2}$ and $g(x)=\frac{e^z-1}{z^2}$

Find the order of the pole and the residue of $$f(z)=\frac{\cos z}{z^2}$$ and $$g(z)=\frac{e^z-1}{z^2}$$ What I said: $f(z)=\frac{\cos z}{z^2}$ has a pole of order $2$ at $z=0$. Then: ...
0
votes
2answers
21 views

Complex numbers homography

I'm struggling to show that if a homography $\frac{az+b}{cz+d}:\mathbb{C} \rightarrow \mathbb{C} $ sends set $\{z:\Im(z)>0\}$ to itself, then $a,b,c,d \in \mathbb{R}$ and $ad-bc>0$ I will be ...
0
votes
1answer
23 views

Find image of complex function: $w(z) = z+i\bar{z}$ defined from $\mathbb{C}$ into $\mathbb{C}$

I found that, $w(\mathbb{C}) = \left \{z \in \mathbb{C}: \Re(z) = \Im(z) \right\}$, is that correct? In such case, does it mean the resulting image set is a line? Any example of a bijection ...
-1
votes
0answers
43 views

De Moivre's theorem and complex number question

I understand how $$((0.5+0.866i)(0.5+0.866i)(0.5+0.866i)(0.5+0.866i)(0.5+0.866i))^{\frac{1}{5}}=0.978-0.208i$$ by using De Moivre's theorem. However, I am confused whether or not the answer could be ...
0
votes
0answers
42 views

Bounds for coefficients of complex polynomial satisfying $|P(z)| \leq 1$ for $|z| \leq 1$

Let $P$ be a complex polynomial: \begin{align} P_n (z) &= a_nz^n+a_{n-1}z^{n-1} + \ldots + a_1z+a_0\ \end{align} In the unit disk the polynomial is bounded, meaning that: \begin{align} |P_n ...
1
vote
1answer
47 views

Show composition of harmonic function and analytic function is harmonic without calculating 2nd derivatives

This is not a duplicate. I realize similar questions to this have been asked, but what I am asking is slightly different: I need to prove the following, arguing by complex differentiability only, and ...
-1
votes
1answer
51 views

Is $(z-i)(z+1)$ a polynomial? [closed]

Suppose I factorise the polynomial $(z+i)(z-i)(z+1)$ and I'm interested in the remainder term, when I factor $(z+i)$ out? Is $(z-i)(z+1)$ a polynomial?
2
votes
1answer
35 views

Find the basis for eigenvalues complex numbers

My Matrix A which acts on $\Bbb{C}^2$: $$\begin{vmatrix} 0&1\\ -18&6 \end{vmatrix}$$ I worked out my eigenvalues: $3 + 3.i, 3-3.i$ Now I need to find the basis for the eigenspace: ...
-1
votes
0answers
8 views

An application of Toeplitz determinant

Given a positive real c between 0 and 2 and d is complex. How to show 4 + Re{(c^2)*d} - |d|^2 - 2(c^2) greater or equal to 0 is equivalent to 2d = c^2 + x((4 - c)^2), for some complex x where x is ...
0
votes
1answer
28 views

How to this Complex problem?

Let $$p(z)=A_n z^n+A_{n-1}z^{n-1}+\dots+A_1z+A_0,$$ then $p(w_1)=0,p(w_2)=0,\dots,p(w_n)=0$. Why $w_1+w_2+\dots +w_n=-\frac{A_{n-1}}{A_n}$? I understand only Real part. But I don't know the ...
1
vote
2answers
55 views

$a$,$b$ and $c$ are roots of the equation $x^3-x^2-x-1=0$

The roots of the equation $x^3-x^2-x-1=0$ are $a$,$b$ and $c$. if $n \gt 21 $ and $n \in \mathbb{N}$ The find the possible values of $$E=\frac{a^n-b^n}{a-b}+\frac{b^n-c^n}{b-c}+\frac{c^n-a^n}{c-a}$$ ...
-4
votes
2answers
26 views

Prove this complex number proposition. [closed]

Prove that $$\left| |z_1|-|z_2|\right| \le|z_1+z_2|,$$ where $z_1,z_2 \in \mathbb{C}$.
3
votes
1answer
39 views

Use an expression for $\frac{\sin(5\theta)}{\sin(\theta)}$ to find the roots of the equation $x^4-3x^2+1=0$ in trigonometric form

Question: Use an expression for $\frac{\sin(5\theta)}{\sin(\theta)}$ , ($\theta \neq k \pi)$ , k an integer to find the roots of the equation $x^4-3x^2+1=0$ in trigonometric form? What ...
1
vote
1answer
22 views

Solution of equation are vertices of polygon

the solution of equation $z^4+4z^3-6z^2-4iz-i =0$ (where $i=\sqrt{-1}$) are the vertices of convex polygon in complex plane.Find the area of the polygon. How should we approach this question? Could ...
5
votes
0answers
78 views

Generalization of FTA

I'm sure that this is not any hypothesis, but following came to my mind when I was reading complex analysis. Consider a function $f(z)=z^n+g(z)$, where $g(z)$ is continuous (not necessary ...
1
vote
1answer
19 views

Solving for complex $a_n$ in a harmonic series

Is there any way to know for what values of $a$ and $b$, the following is true? $$\sum_{n=1}^\infty \frac {(-1)^{n-1}}{n^{a+b*i}} = 0$$
4
votes
4answers
63 views

How do I find the solution to the equation $z^2=-81i$?

This question is from the Powers of Complex Numbers, Precalculus section of KhanAcademy Find the solution to the following equation whose argument is between $90°$ and $180°$ $$z^2=-81i$$ What I ...
2
votes
2answers
31 views

Determine whether $f_n(z)=nz^{\sqrt{n}}(1-z^{\sqrt{n}})$ converges pointwise

Let $$f_n(z)=nz^{\sqrt{n}}(1-z^{\sqrt{n}})$$ on $E:=\{z:|z|<1\}$. Does $f_n(z)$ converge pointwise? I think that $$\lim_{n\to\infty}f_n(z)=0$$ but I am not sure how to formally show that, i.e. ...
1
vote
1answer
46 views

If $f$ is holomorphic, what is the meaning/intuition behind $f_z=f'(z)$ and $f_{\overline z}=0$?

If $f$ is holomorphic then we know the derivative of $f$ with respect to $z$ is defined, i.e., $f'(z)$ exists. But $\overline{z}$ is a different variable, so if we take the derivative of $f$ with ...
0
votes
1answer
30 views

Show that if $w^3=1$, then $1+w+w^2=0$

Find the cube root of unity. Hence, show that if $w^3=1$, then $1+w+w^2=0$. The cube root of unity is $1, -\frac{1}{2}+\frac{\sqrt{3}}{2}$ and $ -\frac{1}{2}-\frac{\sqrt{3}}{2}$. If $w^3=1$ ...
0
votes
1answer
24 views

Prove that $f_n$ does not converge pointwise

If $$f_n=\frac{\sqrt{n}}{z^3}$$ on $E:=\{|z|\leq17\}$, does it converge pointwise? I know that it does not, since as $n\to\infty$, $f_n\to\infty$ independent of $z$, but how do I convey this ...
1
vote
3answers
48 views

Residue of $f(z) = \frac{1}{z-\sin z}$ at $z=0$

My attempt: $$ f(z) = \frac{1}{z-\sin z}$$ $$\frac{1}{z-(z-\frac{z^3}{6}+\frac{z^5}{120}-...)}$$ $$\frac{1}{z(1-(1-\frac{z^2}{6}+\frac{z^4}{120}-...))}$$ $$Res(f(z),0) = \lim_{z \to 0} z \cdot ...
1
vote
2answers
43 views

Confusion about nth root of a number

Let's say we want to find the 6th roots of 64. Then according to the method from my textbook: $z^6=64$ $z^6=64+0i=64[\cos(0)+i\sin(0)]=64cis(0)=64cis(0+2k\pi)$ Then by De Moivre's theorem: ...
2
votes
2answers
18 views

If $z^n+z^{-n}=2\cos(n\theta)$ show that $5z^4-z^3-6z^2-z+5=0 => 10\cos^2(\theta)-\cos(\theta)-8=0$.

If $z^n+z^{-n}=2\cos(n\theta)$ show that $5z^4-z^3-6z^2-z+5=0 => > 10\cos^2(\theta)-\cos(\theta)-8=0$. I can prove $z^n+z^{-n}=2\cos(n\theta)$ with De Moivre's theorem (the sines cancel out ...
0
votes
1answer
29 views

To evaluate square roots of $1+2i$.

Here is as far as I got. First we write $1+2i$ in the polar form which is $\sqrt{5}e^{i\alpha}$ ($\alpha$ is the argument of $1+2i$ which turns out to be $\arctan2$). Therefore the square roots are ...
3
votes
2answers
74 views

What was the motivation for the complex plane?

I've read a bit about the history of the complex numbers, and many seem to credit Caspar Wessel with the idea of associating the complex numbers as points on a 2-dimensional plane (or at least the ...
1
vote
3answers
32 views

Complex Numbers: How do I solve for relation between two complex numbers?

Prove by contradiction that if $w,z\in\mathbb{C}$ such that $|w|\leq 1$ and $w^nz+w^{n-1}z^2+...+wz^n=1$, then $|z|>(1/2)$. Any help on how to approach the problem will be helpful. I don't have ...