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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Complex vector identity

Let $f=(f_1,f_2,f_3)$ be a complex vector. Can we see that $$G:=\frac{2\Im(f_2\bar{f_3})+i2\Im(f_3\bar{f_1})}{|f|^2-2\Im(f_1\bar{f_2})}=\frac{f_3}{f_1-if_2}$$ I tried using $f_j=x_{j,u}-ix_{j,v}$ ...
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1answer
20 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
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2answers
34 views

Complex Number, Quaternions and Octonions [duplicate]

There are complex $\mathbb C$, quaternions $\mathbb H$ and octonions $\mathbb O$. Is there any higher dimensional generalization of them, such in the $\mathbb R^{16}$? Or why do we just study three ...
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1answer
44 views

Real Manifold … Complex Coordinates?

I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 ...
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1answer
20 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
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3answers
58 views

Complex numbers? [duplicate]

There are plenty of questions out there asking what complex numbers mean and I never seem to get any of them. I have a few specific questions i want to ask about complex numbers. 1) what is the ...
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1answer
31 views

Proving relation for square root of complex number

How do I represent $\sqrt{1 + ja}$? I'm trying to show that it's approximately equal to $\pm(1 + \frac{ja}{2})$ when $a \leq 1$.
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2answers
28 views

Ignoring the pole?

I have the integral $$\int^{2 \pi}_0 \frac{z}{z+2} dz$$ where $$|z|=3$$ I parametrise the integral and get $$\int^{2\pi}_0 \frac{9 ie^{2i\theta}}{3e^{i\theta}+2}$$ and this gives the required answer ...
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1answer
24 views

Evaluating a complex integral in punctured plane.

I am trying to evaluate the complex integral $$\int \frac{z}{z+2} dz$$ And $$|z|=3$$ We can see there is a pole at $z=-2$. How do I go about solving this, what is the strategy?
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1answer
27 views

Arithmetic progression with complex common difference?

Suppose we have the following sequence: $$\{0,i,2i,3i,4i,5i\}$$ Can we call this sequence an arithmetic progression with first term $0$ and common difference of $i$ ? Clarification: Here, $i$ is ...
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2answers
33 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
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0answers
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Find a generator for vectorial subspace

S = {$(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0$} $c+d-i(2ia)=0$ $c+d+2a=0$ $c=-d-2a$ $(a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1)$ Is this solution correct?
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3answers
59 views

Why imaginary numbers axis is plotted perpendicular to the real numbers axis?

Negative numbers axis is plotted to the opposite side of the positive real number axis that make sense but i do not understand why imaginary numbers are plotted perpendicular to the real numbers axis. ...
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4answers
244 views

Cauchy integral formula

Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! ...
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0answers
36 views

Complex function

Can anyone give me a hint to approach this question? I haven't done anything like this before so I'm bit confused about what this question is asking. Thank you very much for all your help.
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1answer
35 views

Midpoint of two complex numbers in polar form

Say we have two complex numbers: $re^{i\theta}$ and $se^{i\phi}$ Is there a straightforward way to find the polar form of the midpoint of these two complex numbers? I think I'm correct in saying ...
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2answers
52 views

Help solve ${{z}^{3}}=\overline{z}$ ($z\in \mathbb{C}$) [duplicate]

Me and my friend try to solve $${{z}^{3}}=\overline{z}$$ where $z \in \mathbb{C}$. My way to solve it was: $\operatorname{cin}(\theta )=\cos(\theta)+\sin(\theta)i$ \begin{align} & z=r ...
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3answers
18 views

Electrical Engineering (complex numbers)

Electrical Engineering ($j=i=\sqrt{-1}$): $$H_v(\omega)=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega CR}{1+Rj\omega C}$$ And we know that: $\omega_0=\frac{1}{RC}\Longleftrightarrow ...
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3answers
20 views

Complex plane (Show that triangle is right-angled)

The points $O$,$P$ and $Q$ in the complex plane represent the complex numbers $0+0i$, $4+2i$ and $3-i$ respectively. Find the exact length of $PQ$ and hence, or otherwise, show that triangle $OPQ$ is ...
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1answer
32 views

Complex Number (Angle)

The complex number $z$ is given by $z=-2+2i$ Find the modulus and argument of $z$ Write down the modulus and argument of $\frac{1}{z}$ Show on an Argand diagram the points A,B and C representing the ...
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1answer
29 views

Complex Numbers (Find p and q)

The complex numbers z1 and z2 are given by $$z_1=5+i,z_2=2-3i$$ Determine the values of the real constants $p$ and $q$ such that $$\frac{p+iq+3z_1}{p-iq+3z_2}=2i$$ My attempt, I substitute $z_1$ and ...
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3answers
110 views

Complex Numbers (Modulus)

The complex numbers $z_1$ and $z_2$ are given by $$z_1=5+i,z_2=2-3i$$ Find the modulus of $z_1-z_2$ My attempt, modulus of $z_1-z_2=\sqrt{5^2+1^2}-\sqrt{2^2+3^2}$ $=\sqrt{26}-\sqrt{13}$ ...
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1answer
20 views

Derivatives of real part of function

My physics textbook gives me the complex form equation of simple harmonic motion as: $$z = Ae^{i(\omega _{o}t+\phi )}$$ and then defines $$ x = Re (z) $$ From there they argue that $$ \frac{\partial ...
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1answer
27 views

Explanation of two argument variant for arctan

Can someone please explain why $$\tan^{-1}\left(\frac{y}{x}\right)$$ has the additional conditions based on what the value of x and y are? I'm most specifically interested in the second equation: ...
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3answers
41 views

Cube roots of complex numbers [on hold]

I need help with finding the cube roots of the complex number 27... I know that the obvious answer is three, but what is the less simple method to solving this?
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0answers
13 views

Simplifying complex exponential

Two simplifications from my book that I don't understand, first: 5exp(-j1.571) = - j5 Why does the real part get dropped off? Also: exp(j3.785) = -exp(0.643) Is there a way to directly covert ...
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2answers
38 views

Using Eulers formula

I am trying to figure out how \begin{equation*} e^{i(-1+i\sqrt{3})}=e^{-\sqrt{3}} (cos(1)-i sin(1))?? \end{equation*} I know that Euler's formula states that \begin{equation*} e^{ix} = \cos(x) + i ...
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1answer
23 views

Evaluating complex integral on circle

I am trying to evaluate the integral $$\int \frac{2z-1}{z(z-1)} dz$$ counter clockwise around the circle $$|z|=2$$ First we apply partial fraction decomposition to get $$\int \frac{1}{z}+\int ...
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2answers
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Complex number (Rhombus)

Given that $z_1=1+2i$ and $z_2=\frac{3}{5}+\frac{4}{5}i$, write $z_1z_2$ and $\frac{z_1}{z_2}$ in the form $p+iq$, where $p$ and $q \in R$. In an Argand diagram, the origin O and the points ...
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1answer
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Sum of complex series

After stating the sum I wrote z in polar form and then proceeded to calculate the real part of the sum I stated in the first part. However the working got tedious very soon and I was not able to ...
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1answer
22 views

Argument of Complex Number (Am I wrong?)

I'm given $z=-2+\sqrt{3}i$. So I worked out the argument of $arg(z)=\tan^{-1}(\frac{\sqrt{3}}{-2})$. I got the answer $2.256$rad. But the given answer is $2.45$rad. Am I wrong?
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0answers
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Integrating complex functions over the unit circle

I am trying to evaluate $$\int_c \bar z dz$$ where the contour is the unit circle. I know the limits of $\theta$ is $0 \to 2\pi$ How do I get to the answer of $2\pi i$?
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3answers
66 views

Finding local max of analytic function

Given a function $f=z^2+iz+3-i$. I need to find the the maximum of $|f(z)|$ in the domain $|z|\leq 1$ I know that the maximimum should be on $|z|=1$ but when I tried to put $z=e^{i\theta} $ in the ...
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2answers
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Simplifying Complex numbers

Help me simplify this complex number: Hints are welcome, so that I can see how to move on $$\left(\frac{1+6i}{\sqrt{76}e^{\frac{1}{2}\pi i}}\right)^{2i}$$
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39 views

Complex number $\tan \alpha+i$

Given that $z=\tan \alpha+i$, where $0<\alpha<\frac{1}{2}\pi$ Find $\left |z \right |$. I've never seen this kind of example in my book. Can anyone guide me? Thanks a lot. How to find $arg ...
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2answers
181 views

Square roots of Complex Number. [duplicate]

Calculate, in the form $a+ib$, where $a,b\in \Bbb R$, the square roots of $16-30i$. My attempt with $(a+ib)^2 =16-30i$ makes me get $a^2+b^2=16$ and $2ab=−30$. Is this correct?
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1answer
32 views

Find $\int_c \bar z$ along the parabola $y=x^2$ from $(0,0)$ to $(1,1)$

I know $\bar z=x-iy$ So we have $$\int_c x-iy \,dz$$ when split up gives us $$\int^1_0 x \, dx-i\int^1_0 x^2 \cdot 2x \, dx$$ and then I integrate as usual as usual and I get the result ...
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2answers
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equation of a line into the complex form

So if i am given an equation of a line in complex form for example $Re|(1+i)z| = 0$, I could turn this into its real counter part on the x-y plane and graph it. Is there a way to go in the other ...
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0answers
27 views

complex variable inequality

Let $B$ and $C$ be nonegative real numbers and $A$ a complex number. Suppose that $$ 0\leq B-2Re(\overline{\lambda}A) + |\lambda|^2 C \ \forall \ \lambda \in \mathbb{C} $$ Conclude that $|A|^2 \leq ...
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1answer
25 views

How to find complex coordinates of a square?

If one coordinate is given by: $z_{1}=\frac{3}{2}+\frac{3}{2}i$ and $Re(z_{2})=6,Re(z_{4})=1$. How to find $z_{2},z_{3},z_{4}$ so that $z_{1}z_{2}z_{3}z_{4}$ forms a square in the first quadrant? ...
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Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
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How is the second part of a dual number called?

A complex number $a + bi$ has a real part $a$ and an imaginary part $b$. But, what about dual numbers $u + v\epsilon $? I have seen the non-real part $v$ been called the infinitesimal part. Is this a ...
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2answers
44 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?
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3answers
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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 + ...
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7answers
95 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 ...
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1answer
20 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 ...
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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 ...
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1answer
16 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\, ...
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1answer
17 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 ...
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2answers
29 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