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

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4
votes
1answer
33 views

Is this a sound demonstration of Euler's identity?

Richard Feynman referred to Euler's Identity, $e^{i\pi} + 1 = 0$ as a "jewel." I'm trying to demonstrate this jewel without recourse to a Taylor series. Given $z = cos\theta + i sin\theta\; |\;|z| = ...
0
votes
1answer
15 views

Probability distributions for complex numbers

Aside from the complex normal distribution (which is really just a bivariate normal applied to the real and imaginary parts), are there any other distributions involving complex numbers that have ...
3
votes
4answers
33 views

Solving complex equation ${{\left( z+2\bar{z} \right)}^{3}}=1$

Hey i am stuck tring to solve : $${{\left( z+2\bar{z} \right)}^{3}}=1$$ I used Binomial theorem to expand the equation to : $${{z}^{3}}+6{{z}^{2}}\bar{z}+12z{{\bar{z}}^{2}}+8{{\bar{z}}^{3}}=1$$ and i ...
0
votes
4answers
46 views

Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$

Show that $ \quad \quad z^5-1=(z-1)(z^2+2z\cos{\dfrac{\pi}{5}}+1)(z^2-2z\cos{\dfrac{2\pi}{5}}+1)$ and deduce from this the closed formulas for $\cos{\dfrac{\pi}{5}}$ and $\cos{\dfrac{2\pi}{5}}$. ...
0
votes
1answer
33 views

Need help with proof with absolute value and complex numbers. [duplicate]

Had some trouble trying to prove the following problem. Prove that if $|z| < 1$ and $|w| < 1$, then $$ \frac{|z-w|}{|1-\overline{z}w|} < 1 $$ Would appreciate some help.
0
votes
1answer
16 views

A unique expression for a unitary complex.

In an attempt to show that a unitary complex $z$ must be of the form $z=\cos \alpha + i \sin \alpha $ for some $\alpha \in \mathbb R$ we are led to this situation: Let $x$ and $y$ be reals such that ...
-2
votes
0answers
43 views

Complex numbers - What am I missing? [on hold]

I got stuck on this problem, I hope someone could help me solve this please. so this is the equation: $$(1+i)z^2+(19+3i)z+54-28i = 0$$ Thanks:)
1
vote
0answers
19 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
1
vote
3answers
490 views

Complex Numbers - Weird Equation

I'm not sure if it is a error typing in the book or just something I can't solve hope you could help me, so here it is: $$z \bar z+z^2 = \bar {\bar z} - z +2$$ This is the same way they write it in ...
2
votes
4answers
56 views

How do you evaluate an expression if the variables are complex numbers?

Let $x$, $y$, and $z$ be complex numbers. if $x + y + z = 2$, $x^2 + y^2 + z^2 = 3$, and $xyz = 4$, then evaluate the expression $$\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{zx+y-1}$$
-1
votes
0answers
14 views

Can a harmonic function exist which doesn't have a harmonic conjugate. [on hold]

Will partial derivatives of u(x,y) always be harmonic if u(x,y) is harmonic?
4
votes
4answers
61 views

Find $z$ when $z^4=-i$?

Consider $z^4=-i$, find $z$. I'd recall the fact that $z^n=r^n(\cos(n\theta)+(i\sin(n\theta))$ $\implies z^4=|z^4|(\cos(4\theta)+(i\sin(4\theta))$ $|z^4|=\sqrt{(-1)^2}=1$ $\implies ...
0
votes
1answer
19 views

what is the image of $z/|z|$ where z is complex

what is the image of $z/|z|$ where z is complex? I know it is the unit circle . and that in R it means the sign (1, -1) but what is it on the complex numbers? is it $\Re_+$ or is it $S^1$? also - ...
0
votes
0answers
17 views

Can I determine complex differentiability by differentiating wrt to z?

If I have a function in terms of $z\in C$ and need to determine the points where it is differentiable, can I simply find the derivative wrt z and see where it is defined? I know that one solution is ...
0
votes
0answers
40 views

Finding the eigenvalues of this matrix.

Suppose I wanted to find the eigenvalues of the two by two matrix A: $$ A=\pmatrix{i&1\\0&1} $$ We see that $A(x,y)=(xi+y,y)$ I see that $\lambda = i$ is an eigen value, with the ...
5
votes
1answer
88 views

Proving $(1-x)\cdot (1-x^2)\cdots(1-x^{n-1})=n$ if $x^n=1$ and $x\neq 1$ [on hold]

If we have a equation $x^n=1$, then how can we prove $$(1-x)\cdot (1-x^2)\cdots (1-x^{n-1})=n $$ when $x$ is not $1$? I know that $x= e^{(2\pi + 2k\pi)/n}$ and we can get different value of $x$ when ...
0
votes
3answers
30 views

If $|z_1 - z_2| = |z_1 + z_2|$, then $|\arg z_1 - \arg z_2 |= \frac{\pi}{2} $ or $\frac{3\pi}{2}$

If for $z_1, z_2\in \Bbb C $, $|z_1 - z_2| = |z_1 + z_2|$, then we have to prove $|\arg z_1 - \arg z_2| = \frac{\pi}{2} $ or $\frac{3\pi}{2}$. I have seen similar type question here If $|z_1 - ...
0
votes
2answers
34 views

Terminology for complex numbers

Let $c=x+iy$ be a complex number (with $x\ne0, y\ne0$). Then $x-iy$ is the conjugate of this number. Is there any term (or use) for the number $-x+iy$?
0
votes
3answers
41 views

Principal $n$th Root

This is a simple question, but I am having a glitch somewhere: What is the principal $n$th root of $(-9)^{1/2}$? I keep getting $3$ and the book has $3i$. I know it has to do with the $-9$, but I ...
1
vote
1answer
21 views

Riesz representative of gradient of $f(u) = u^*u$ in different inner products

This is a seeming "paradox" that has been bothering me for some time, as it (or other situations like it) show up often when computing gradients for numerical optimization on complex vector spaces. ...
2
votes
2answers
29 views

$(x_n,y_n) \in \{(x,y)\mid y=kx\}, n=1,2,3,4\Leftrightarrow$ $\frac{z_1-z_3}{z_2-z_3} \div \frac{z_1-z_4}{z_2-z_4} \in \mathbb{R}, z_n = x_n+iy_n$

I need to prove that 4 points of the complex plane lie on a same line or a same circle if and only the following is right for corresponding complex numbers: $$\frac{z_1-z_3}{z_2-z_3} \div ...
0
votes
6answers
65 views

Find solution to complex equation

Find all the complex solutions to the equation $$iz^2+(3-i)z-(1+2i)=0$$ I've tried to solve this equation with two different approaches but in both cases I couldn't arrive to anything. 1) If ...
1
vote
0answers
37 views

Possible proof strategy for Sendov conjecture?

Sendov's conjecture says that if all roots of a polynomial lie within the unit disk, then for every root, there exists a critical point at a distance at most one from the root. I read that Sendov ...
-1
votes
1answer
30 views

Find all the values of the following [on hold]

I got tasked to "find all the values of the following", but i struggle to comprehend the question. What values am i asked to find? (We are working with complex numbers) a) $1^{\frac{1}{5}}$ b) $(1 - ...
1
vote
4answers
135 views

imaginary number $i$ equals $-6/3.4641$? [duplicate]

$$-4^3 = -64$$ so the third root of $-64$ should be $-4$ than. $$\sqrt[3]{-64} = -4$$ But if you calculate the third root of -64 with WolframAlpha( ...
0
votes
1answer
25 views

Taylor series on complex analysis

Suppose that, I have $\sum_{n=1}^\infty (z^n)/n$. Now clearly for the open disk $|z|<1$, above series converges. But if I consider $|z|=1$, then clearly for $z=1$, above series diverges. How do I ...
0
votes
1answer
18 views

Creating a Hermitian matrix that is also positive semi-definite

Given some measurements on empirical data (in the form of a multigraph with two weighted edges between every pair of vertices), I would like to place the measurements in a Hermitian matrix that also ...
2
votes
1answer
61 views

Are there any solution for a,b,c,d such that $(a+bi)^{n}+(c+di)^{n}=2i$

Are there any solution for a,b,c,d such that $(a+bi)^{n}+(c+di)^{n}=2i$. With a,b,c,d,n are positive integer numbers and $a+bi, c+di$ are complex numbers . I just have started learning about comlex ...
0
votes
1answer
20 views

Show that a set of polynomials are linearly independent in the complex space

I have been trying the solve the following question without any success: Let $\lambda_1, \lambda_2, \lambda_3$ be three distinct complex numbers and define the polynomials $m(\lambda), m_1(\lambda), ...
0
votes
1answer
24 views

How do you show that f(z)=z conjugate isn't linear?

let $x_1= a+ib,x_2= c+id,k=$scalar $f(x_1,x_2)=f(x_1) + f(x_2)$ $f(a+ib + c + id)=(a+c)-i(b+d)$ $f(a+ib)+f(c+id)=(a+c) - i(b+d)$ $f(kx_1)=kf(x_1)$ $f(k(a+ib))= k(a-ib)$ $kf(x_1)=k(a-ib)$ Looks ...
0
votes
0answers
18 views

What are the loci of points z which satisfy the following relations?

a) |Z-Z1|=|Z-Z2| b) 0< Re(iZ)<1 c) |z|=ReZ+1 d) Im((Z-Z1)/(Z-Z2))=0 The professor did not explain loci in class and the text does not have any examples, so I am completely lost.
1
vote
1answer
33 views

Powers of complex numbers

Prove that $\left(\sqrt{3}-i\right)^n = 2^n \left(\cos(n\pi/6)-\sin(n\pi/6)\right)$ $(1+\cos\alpha+i\sin\alpha)^n = 2^n\cos^n(\alpha/2)(\cos(n\alpha/2)+i\sin(n\alpha/2))$ I am completely lost with ...
0
votes
1answer
27 views

Complex conjugate of $z$ as a different variable

Can a complex conjugate be represented by a different letter than $z$? As in: Let $y$ be a complex number satisfying $|y|<1$. Find the set of all complex numbers $z$ satisfying ...
0
votes
2answers
52 views

(complex analysis) Prove that: $\arg ((z_3-z_2)/(z_3-z_1)) = 1/2 \arg z_2/z_1$

If $|z_1|=|z_2|=|z_3|$ Urgent help needed. I have used: $z_1=x_1+\mathrm iy_1,z_2=x_2+\mathrm iy_2,z_3=x_3+\mathrm iy_3$ and obtained $$\arg\frac{z_3-z_2}{z_3-z_1} = \arctan ...
1
vote
1answer
20 views

Rewriting a trig function into a sum of exponential functions

Rewrite the function $2 + 4\sin(\pi t + \frac{\pi}{6})$ into a sum of exponential functions. By that I mean using Euler's formula $\sin(x) = \dfrac{e^{i\pi x} - e^{-i\pi x}}{2i}$. If it wasn't for ...
1
vote
2answers
34 views

Find the argument of $\dfrac{(3-2i)(1-i)}{(2+i)^2}$

As the header suggests, I am supposed to find the argument for the complex number $\dfrac{(3-2i)(1-i)}{(2+i)^2}$ This is how I've tried: Approach 1: Calculate the arguments by factoring out the ...
1
vote
1answer
32 views

Find the analytic function

$f(z)=1 $ satisfies the condition Using Identity Theorem $f(z)=1$ can be only function that satisfies this. so option (b) is NOT true. Am I on correct path?
0
votes
0answers
11 views

On charge conjugation of Dirac spinor

Suppose we have Weyl spinor $\psi_{a}$, which transforms under irreducible representation $\left( \frac{1}{2}, 0\right)$ of the Lorentz group, $$ \psi_{a} \to (T(g))_{a}^{\ b}\psi_{b}, $$and complex ...
0
votes
1answer
23 views

why $f(z) = z^{(3/2)}$ does not have derivative at z = 0 in complex plane.

it seems that the $f'(z) = z^{(1/2)}$ means that this function has derivative for every complex value. But why $f(z) = z^{(3/2)}$ does not have derivative at z = 0
1
vote
1answer
36 views

Prove that $\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$

As the title suggests, what is required to prove is that $$\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$$ I was looking back through my ...
0
votes
0answers
12 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
1
vote
1answer
34 views

There exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$. Show that $f$ can be extended analytic on $\Bbb C$.

(a) Suppose that $f$ is analytic on the open unit disk $\{z: |z|<1 \}$ and there exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$ for all $k \geq 0$. Show that $f$ can be extended analytic on ...
0
votes
2answers
68 views

Why is Euler's formula a definition?

Even though there are proofs for Euler's formula for complex exponentials (see wikipedia for instance), it is mentioned as a "definition" in most textbooks. Why is that? My understanding is that a ...
0
votes
3answers
31 views

Let $A$ be a complex number and $B$ be a real number. Prove that $\mid z^2\mid+Re(Az)+B=0$ can only have a solution iff $\mid A^2 \mid \ge 4B$.

Been stumped on this question for a while. I tried letting $z=\mid z \mid \cdot e^{i \alpha}$ and $A=\mid A \mid \cdot e^{i\beta}$ -- assuming that $\alpha$ and $\beta$ were the arguments of $z$ and ...
3
votes
2answers
42 views

Does the identity ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ given in my text hold?

In my text book I saw that ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ But when I tried deriving it myself I got this: $${|\cosh z|}^2={\cos}^2y+{\sinh}^2x$$ See my working below: $$\cosh ...
0
votes
1answer
26 views

Find the location of the center and the radius of the following circle: [closed]

Find the location of the center and the radius of the following circle: $$ \left| \ \frac{z-1}{z+1} \ \right| \ = \ 3 \ \ . $$ $ \ z \ $ is a complex number. Thanks in advance!
4
votes
1answer
85 views

Find a convergent solution for $a$

Find a value for $a$ in which the following sum converges. $$a+a!+(a!)!+((a!)!)!+\cdots$$ I know that there are no solutions if you only look at $a\in \Bbb{R}$, but are there any solutions if you ...
0
votes
0answers
30 views

How can we represent complex numbers in 2-d plane(i.e. complex plane) if there is no ordinal relationship between them? [closed]

If there is no ordinal relationship(2i is not greater or equal or lesser than i i.e there is no order relation between i and 2i) in complex numbers then why are they represented in ordinal manner in ...
2
votes
1answer
22 views

A set $I$ of isolated complex numbers such that $[0,1]\subset\{Re(z):z\in I\}$

Is there a set $I$ of isolated complex numbers, such that $$[0,1]\subset\{Re(z):z\in I\},$$ where $Re(z)$ is the real part of the complex number $z$.
0
votes
0answers
35 views

about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$

I am a little confused about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$. From other answers (Is a complex vector space closed under complex ...