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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2
votes
3answers
51 views

Show that $\frac{1+\cos 5x+i\sin 5x}{1+\cos 5x-i\sin 5x}=\cos 5x+i\sin 5x$

$$\frac{1+\cos 5x+i\sin 5x}{1+\cos 5x-i\sin 5x}=\cos 5x+i\sin 5x$$ When I attempted this I first tried multiplying top and bottom of the LHS by the complex conjugate of what's on the bottom, $1+\cos ...
1
vote
3answers
49 views

What's the difference between $|z|^2$ and $z^2$, where $z$ is a complex number?

I know that $|z|^2=zz^*$ but what is $z^2$? Is it simply $z^2=(a+ib)^2$?
6
votes
3answers
699 views

Having trouble understanding generalized complex numbers

I'm reading a paper on the generalized complex numbers, but I have trouble in some of its fundamental properties. I have searched wiki but it left me none the wiser. Please see the image below, in ...
1
vote
1answer
50 views

Prove $(Imz)^2$ is not holomorphic

Prove that the function $(Imz)^2$ is not homolomorphic in any subset of $\mathbb{C}$ I know I need to use Cauchy Riemann's equation where $\frac{dv}{dy} = \frac{du}{dx}$ and $\frac{dv}{dx} = ...
3
votes
3answers
64 views

Is the set $S=\left\{\left(z_1,z_2\right)\in \mathbb C\times \mathbb C:z_1^2+z_2^2=1\right\}.$ compact?

Consider the set $$S=\left\{\left(z_1,z_2\right)\in \mathbb C\times \mathbb C:z_1^2+z_2^2=1\right\}.$$ Is this set compact in $\mathbb C^2$ ? As $\mathbb C^2$ is a finite dimensional space so a ...
0
votes
1answer
68 views

Computing a contour integral of a function that is not analytic inside the contour

I'm wondering if there is another way to calculate the contour integral of $\int(\tan(z/2)/(z-1))$ in the square w/ sides $Re(z)=+/-2$, $Im(z)=+/- 2$ other than using the residue theorem. The cauchy ...
3
votes
2answers
25 views

Prove that $S_{n}(x) = \sum_{k=-n}^{n}c_{k}e^{ikx}$

The problem is to prove that every trigonometric sum of the form $$S_{n}(x) := \frac{1}{2}a_{0} + \sum_{k=1}^{n}(a_{k}\cos kx + b_{k} \sin kx)$$ can be expressed as $$S_{n}(x) = ...
0
votes
1answer
75 views

How did i go? Find the exact value of … $\frac{z^{43}}{w^{87}}-\frac{\bar{z}^{23}}{w^{41}}$

$$\frac{z^{43}}{w^{87}}-\frac{\bar{z}^{23}}{w^{41}}$$ $ z=1-\sqrt{3}i = 2e^{\frac{5pi}{3}i} \\ \bar{z} = 1+\sqrt{3}i = 2e^{\frac{pi}{3}i} \\ w = 1+i=\sqrt{2}e^{\frac{pi}{4}i}$ Then by de moivre's ...
2
votes
0answers
48 views

compute the complex-valued integral for the branch cut

Let $C$ be the circle of radius $2$ centered at origin. Let $f(z)$ be the branch cut of the function $z^{2−i}$ on the domain $−π < θ < π$. Compute the integral $$ \int_C f(z) dz$$ My attempt: ...
2
votes
2answers
61 views

Taylor series for the function $f(z) = \frac{1}{(z-5)(z-7)}$ on a disc centered at point $z_0=3$

I started by expressing the function as sum of two fractions using partial fraction decomposition to get $\frac{-1}{2(z-5)} + \frac{1}{2(z-7)}$ However I could only then end up writing that as the ...
0
votes
2answers
296 views

Linear Independence of a set of Complex Vectors

I am trying to understand how to determine the linear dependence/independence of a set of complex vectors. I know the process is the same as determining linear dependence/independence of a set of ...
2
votes
5answers
124 views

I am getting $i^{-1}=\pm i$.

I am trying to find $i^{-1}$. I already know that the answer is $-i$, but I can't figure out a way to determine that using math. This is what I am doing: $$i^{-1}$$ $$\frac1i$$ ...
0
votes
0answers
28 views

image of $|z|= \pi\ $ under $ z \mapsto\ e^z\ $

I have been asked to sketch the image of $|z|= \pi\ $ under the mapping $ z \mapsto\ e^z\ $ ; I think I should use the identity that $ z = |z| e^{i\theta}$ , and replace z in $ e^z $ with this ...
1
vote
3answers
47 views

Number of solutions for equations with complex variables

A question about the number of solutions for the following equation: $$z^2+(1-i)z-3i=0$$ So the solutions are: $$z_{1,2}=\frac{-1+i \pm \sqrt{10i} }{2}$$ But $\sqrt{10i}$ has two options with ...
0
votes
1answer
27 views

Prove that the matrix is hermitian

Prove that if $a*b\ge0$ and $|a|=|b|$ then the matrix below is Hermitian. Where $a,b are possibly complex numbers. \begin{bmatrix} 1&a&0\\b&1&0\\0&0&1 \end{bmatrix} I know ...
1
vote
1answer
36 views

Prove that $\vert z_1-z_2\vert$ is more than or equal to $\vert z_1 \vert - \vert z_2 \vert$

Prove that $\vert z_1 - z_2 \vert \geq \vert z_1\vert - \vert z_2 \vert$ where $z_1,z_2$ are complex numbers. I know that you have to use the triangle inequality for say $\vert z_1+z_2 \vert \leq ...
3
votes
1answer
46 views

The series $\sum\limits_{n\ge1}n^{-z}$ converges locally normally

Show that the series $\sum\limits_{n\ge1}n^{-z}$ converges locally normally on the half plane $\{z:\text{Re}(z)>1\}$ $\displaystyle ...
2
votes
1answer
30 views

Manipulation of matrix of complex numbers

$A \in M_{m \times 1}(\Bbb C),\ B \in M_{m \times n}(\Bbb C),\ C \in M_{n \times 1}(\Bbb C)$. Now when I do matrix multiplication $A^\dagger BC$ where $A^\dagger$ represents the tranpose conjugate of ...
1
vote
1answer
49 views

Prove the result of multiplying a complex number by (1 + i)

I know that if I multiply a complex number that has the form a + bi by the number (1 + i), this will rotate the vector that corresponds to the complex number by 45 degrees in the counterclockwise ...
1
vote
1answer
2k views

How to row reduce a matrix with complex entries?

I have been doing some practice questions for university, and one of them is regarding row reducing a complex matrix. From what I can work out, I think (i could very well be wrong) that the first ...
1
vote
3answers
61 views

Operations and Identities [duplicate]

We have the binary operation addition on numbers. It has an additive identity ( 0 ) and it is commutative. Multiplication is simply repeated addition. It is a binary operation on numbers. Its ...
3
votes
3answers
58 views

Finding the roots of $(1 + i)^{\frac{1}{4}}$

The professor says that the $n = 4$ roots of this are in the form: $\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n})$, where $k = 0, 1, 2, 3$. So to find $\theta$, we find the $r = ...
1
vote
1answer
41 views

Prove: there is a unique pair of integer roots of unity which differ in real part by $1$.

I saw the following lemma somewhere, and I hope I did not misread it: If $z_1$ and $z_2$ are $n$th and $m$th roots of unity respectively ($n,m$ positive integers possible equal), and the real part of ...
1
vote
3answers
65 views

Quotient ring of complex polynomials and ideal domain

Let f(X) = X^2 − 2X + 5 ∈ C[X] and the ideal generated by f(X) be I = f(X)C[X]. (where C(X) is the set of complex polynomials) Prove that the quotient ring C[X]/I is not an integral domain. Since ...
0
votes
3answers
62 views

Solve the equation

Solve the equation $$2{z^2} + 2z + 2iz - 5i = 0 $$ I used the quadratic formula and I have simplified it down to $$\frac{{ - (1 + i) \pm \sqrt {11i} }}{2}$$ Is this correct? Could it be simplified ...
0
votes
1answer
63 views

Defining a rectangular prism using a formula and complex numbers.

I recently read that a line can be defined using the formula $$ A = O + dL $$ where $A$ represents any point on the line, $O$ represents the vector origin of the line, $L$ represents the direction ...
1
vote
1answer
40 views

Proof regarding complex numbers - how to continue?

Let $z,w\in\mathbb{C}$ and $|z|,|w|<1$. Show that $\displaystyle \left|\frac{z-w}{1-z\bar{w}}\right|<1$. My try: ...
3
votes
2answers
87 views

Question about open mapping theorem

Let, $f:\Omega\to \mathbb C$ be a non constant anlytic function on an open set $\Omega \subset \mathbb C$.For $r>0$ let $\mathbb D_r=\{z\in \mathbb C:|z|<r\}$ and let $\bar{ \mathbb D_r}$ be its ...
1
vote
1answer
45 views

How to define complex powers of $0$?

I'm studying Complex Analysis, and I've seen the definition of the set-valued power function as follows Let $z,w \in \mathbb{C}$, then $z^{w} \equiv \exp(w\log z)$. If I recall correctly. Now it ...
0
votes
1answer
19 views

Complex conjugate of the following expression

suppose that we have f is a scalar. And we have the expression $H=Re(f)+Im(f)$. If I want to take the complex conjugate of $H$, does this become $\bar{H}=Re(f)-Im(f)$ or this doesn't make sense?
0
votes
1answer
38 views

Given the entire function $f$, prove that $f(z)=u(x)+\textrm{i}v(y)$ is a polynomial of degree one

So i have written out the Cauchy Riemann equations and have seen that $u_y=0=v_x$. I am trying to think of the relation of these partial derivatives but I'm not so sure how to word my thoughts. I ...
0
votes
3answers
82 views

How to come up with the equation of a line connecting two complex points?

If we have a square with the vertices at the points $0,1,1+i,$ and $i$, how can we come up with equations to represent these lines? For example what is the equation of the line connecting $1+i$ and ...
1
vote
0answers
21 views

Prove that the line integral on $\gamma$ of $z^k dz$=0 for any integer $k\neq 1$ and $\gamma: z(t)=Re^{it}$ $0\leq t\leq 2\pi$

Instead of evaluating it directly I have to show the following: 1) $z^k$ is a derivative of function holomorphic on $\gamma$ and 2) parametrizing $\gamma$ I think I got the first part in which all ...
0
votes
1answer
32 views

Evaluate the line integral of $\frac{1}{z-a}$ over $\gamma$, where $\gamma=a+Re^{it}$, $0\leq t\leq 2\pi$ and a is a complex number

I am having some trouble trying to understand how to deal with the gamma and in general how to fit this into the typical formula. Can anyone help me here out suggest some better way to visualize it?
1
vote
1answer
27 views

Evaluate the line integral over $\gamma$ of |z-1||dz| where $\gamma (t)=e^{it}$, $0\leq t\leq 2\pi$

So far I can figure out that f(z)=|z-1| but I am having trouble parametrizing f(z) to get z(t) along with its derivative and am trying to understand what to do with the $\gamma (t)=e^{it}$. can ...
-1
votes
2answers
40 views

Why does $z=(-i)^{1/2}$ imply $z=e^{(-\pi/2+2n\pi)i}$?

$z $ is a complex number; why does $z=-i^{1/2}$ imply $z=e^{(-\pi/2+2n\pi)i}$? In my textbook this is written without explanation; why is this true? (It says $z^2=-i=exp(i(-\pi/2+2n\pi))$)
0
votes
2answers
105 views

If $f$ is an entire function then what about the set $S=\{Ref(z)+Imf(z) :z\in D\}.$

Let, $f$ be an entire function on $\mathbb C$ and let $D$ be a bounded open subset of $\mathbb C$. Let, $$S=\{Ref(z)+Imf(z) :z\in D\}.$$ Which of the following(s) is(/are) necessarily true ? (a) $S$ ...
0
votes
3answers
123 views

Complex Numbers: Expressing a point on argand diagram

In an Argand diagram, the loci $$ \arg(z-2i) =\pi/6 \quad \land \quad |z-3|=|z-3i| $$ intersect at the point $P$. Express the complex number represented by $P$ in the form $re^{i \alpha}$, where $e$ ...
-2
votes
4answers
93 views

Why do texts frequently define $\mathbf {i}$?

Often when I see a formula containing $\mathbf {i}$, it will be accompanied by the definition $\mathbf {i^2 = -1}$. Why don't we just assume that most students of advanced math know what $\mathbf {i}$ ...
-1
votes
1answer
97 views

Prove that a function is not holomorphic

Prove that the function $(Im z)^2$ is not holomorphic is any open subset of $C$. Please help!
-1
votes
1answer
110 views

Show that the complex function is non-holomorphic everywhere…

Can someone help me with this question: Show that complex function $f(z) = (z^2)*\overline{z}$ is non-holomorphic everywhere.
1
vote
0answers
45 views

Factorising Complex Polynomial with Complex Coefficients

I have tried to factorise the polynomial in question 19 by using the factor theorem to find other factors, however this has been unsuccessful thus far. Seeing as the conjugate root theorem does not ...
1
vote
2answers
197 views

Solve $\arg(-1/z)=-2\pi/3$ and $|1-\frac2z|=1$

$\arg(\frac{-1}z)=\frac{-2\pi}3$ what does this mean? how to i get the $\arg(z)$ from this. I'm thinking of reciprocals. $\left|1-\frac2z\right|=1$ how do i solve for this as well.. i'm confused ...
0
votes
1answer
20 views

Cartesian to polar coordinates, complex numbers

If $$z=re^{i\theta}$$ write $$f(z)=z+\frac{1}{z}$$ as $$f(z)=u(r,\theta)+iv(r,\theta)$$What i did is $$z=re^{i\theta}=r(\cos\theta+i\sin\theta)\space and\space f(z)=z+z^{-1}\space so$$ ...
3
votes
3answers
41 views

Demonstration with complex number

How can I show that $$\overline{e^{i\theta}}=e^{-i\theta}$$ I know that if z is a complex number so $$\overline{e^z}=e^\bar{z}$$ But i don't understand how to show this result.
1
vote
1answer
50 views

Order relation of complex numbers

Show what order relations apply: Set $X = \mathbb{C}$. $(z_1,z_2) \Leftrightarrow Re(z_1) \leq Re(z_2)$ "($z_1$ in relation to $z_2$) is equivalent to ((the real part of $z_1$) $\leq$ ...
4
votes
1answer
182 views

What is the difference between Quaternions and Bicomplex Numbers?

So, I know Quaternions are basically 4 dimensional Complex numbers, and the dimensions can double forever to Octonions, Sedinions, etc. I recently heard about bicomplex numbers, which are also sort of ...
-3
votes
2answers
87 views

How to prove: When the product of two complex numbers is a real number, the complex numbers are proportional to each other's conjugates. [closed]

Looking for a solution to the problem: Given $ z_1, z_2 \in \mathbb{C}$ and $\mathbf(z_1)\cdot(z_2) \in \mathbb{R}$, prove $ z_1 = p\cdot\overline z_2$ for any $ p\in\mathbb{R} $.
1
vote
1answer
50 views

Show that $\sum_{n=1}^{\infty}n^{i(z^2+a)}$ represents an analytic function.

Let, $a\in \mathbb R$ be fixed. Find the set of $z\in \mathbb C$ for which $$\sum_{n=1}^{\infty}n^{i(z^2+a)}$$ represents an analytic function. I know that if the radius of convergence of a power ...
6
votes
3answers
219 views

Does there exist an analytic function $f$ such satisfying the following three conditions?

Does there exist an analytic function $f:\{z\in \mathbb C:|z|<1\}\to \{z\in \mathbb C:|z|<1\} $ such that, $f(0)=1/2$ , $f(1/2)=1/3$ , $f(1/3)=1/4$ ? I tried through the Schwarz-Pick lemma ...