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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75 views

Pre-Calculus Complex Number

My younger cousin asked for help on his math homework and I don't remember doing this, can anyone help please? The denominator of $w$ has $z^*+1$ where the $^*$ means to negate the $z$ term so $z^*= ...
3
votes
3answers
69 views

Closed form of product of complex numbers [duplicate]

I'm stuck in a proof where I want to get a closed form of something. This is the last thing I need to complete my proof: Apparently for small $n\geq2$, the product $\prod\limits_{k=1}^{n-1} ...
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3answers
269 views

Adding complex exponentials

Can somebody please explain $$e^{-\frac{3}{4}\pi i}+e^{-\frac{9}{4}\pi i}+e^{-\frac{15}{4}\pi i}+e^{-\frac{21}{4}\pi i}=0$$ WolframAlpha Computation.
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2answers
27 views

Lower bound this expression

The following expression is a logarithmic expression I am trying to put a lower bound on. Assuming $x,y$ are complex variables. $$F=\log \left( 1 + \big||x|-y\big|^2\right) $$ where the notation |.| ...
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1answer
71 views

Solve $z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$ [duplicate]

Solve $$z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$$ I have never dealt with equations with complex numbers in them so this is interesting; first Ill expand. $$ \implies z^3 + 5z^2- 5iz + 9z + 10 - 10i = ...
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1answer
223 views

Integrate $\int_C{\tan{z}\ dz}; C: y=x^2$ (complex numbers)

Integrate $$\int_C{\tan{z}\ dz}$$ $C$ is the parabola arc $y=x^2$ that connects the points $z=0$ and $z=1+i$. This is what I've done so far: I know that $\tan{z}=\dfrac{\sin{z}}{\cos{z}}$ And ...
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0answers
108 views

Singularities and trigonometric functions

For $f(z)=tan(z)/z$, I have found the singularities to be $z=0, z=\pi/2+2k\pi, z=3\pi/2+2k\pi$. k is an integer. I am trying to find the removable singularities. I have shown z=0 is a removable ...
3
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1answer
131 views

Prove $1! + 2! + 3! + \ldots + n! =y^3$ has only one solution in the set of natural numbers?

I actually know that the above equation is true for $n=1$ and $y=1$ but am unable to prove it for the entire set of natural numbers. Can anyone please help me solve this in a simple way?
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1answer
43 views

complex coordinates of perpendicular chords on unit circle

I am faced with the following problem.. Consider three points $A (a), B (b), C(c)$ on the unit circle $|a|= |b|= |c|=1$. Find the complex coordinates of the point $D (d)$, where $D$ also lies on the ...
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0answers
77 views

complex and decimal tetration

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...
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1answer
55 views

twisted gaussian integers; complex plane with a different basis

I'm trying to understand a kind of twisted form of Gaussian integers. They are defined via $$ w = e^{i \frac{2}{3} \pi}\\ R = \{ m + nw \mid m,n \in \mathbb{Z} \}$$ I tried to picture them by using ...
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4answers
129 views

Show that $i^m + i^{m+1} + i^{m+2} + i^{m+3} = 0$ for all $m ∈ \mathbb N$

Here is the long answer I have come up with so far. m is a natural number If m is divided by 4 :- Let n be the quotient and r be the remainder Then m = 4n+r where 0<=r<4 im = i(4n + r) ...
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0answers
26 views

Action of Möbius Group

By a circle in $\mathbb{C}\cup \{\infty\}$, we mean a circle in plane $\mathbb{C}$ or a straight in $\mathbb{C}$ union with $\infty$. Given two circles in $\mathbb{C}\cup \{ \infty\}$, does there ...
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2answers
65 views

Complex function with real image

Let $f:C \subset \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function and $C$ a connected subset. I want to prove that if $f(z)$ is real for all $z \in C$, then $f$ is constant. Write $f$ as a ...
4
votes
4answers
108 views

Factor the polynomial $z^5 + 32$ in real factors

The question that I have trouble solving is the following: Factor the polynomial $z^5 + 32$ in real factors. The answer should not use trigonometric functions. (Hint: you are allowed to use the fact ...
1
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1answer
64 views

What properties do you lose when you extend your number set? [duplicate]

So in $\mathbb{R}$ and $\mathbb{C}$ you have both associative and commutative property, but as you extend to $\mathbb{H}$ you lose the commutative property, and $\mathbb{O}$ loses the associativity. ...
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1answer
37 views

Solving complex polynomials

I am struggling to solve $z^4+a^4=0$ I believe I need to use De Moivre's theorem, but not sure how to get it to work!
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7answers
225 views

why is $\sqrt{-1} = i$ and not $\pm i$? [duplicate]

this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here ...
1
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1answer
53 views

Does all algebraic developments also work work with complex numbers?

I stumbled on this simple question in my mathbook: Show that the equality: $$\left( \frac{z(z+1)}{2} \right)^2 + (z+1)^3 = \left( \frac{(z+1)(z+2)}{2} \right)^2$$ for all z $\in \mathbb{C}$ Now ...
4
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1answer
176 views

Can someone prove why $\sqrt{ab}=\sqrt{a}\sqrt{b}$ is only valid when a and b are positive?

I have seen many people say that a and b can't be positive for example in this false proof : $$1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1} \sqrt{-1} = i^2 = -1$$ Trust me, I understand that $1\neq -1$ and ...
2
votes
1answer
168 views

if $\alpha$ $\beta$ and $\gamma$ are the roots of a equation than find the value of .

if $\alpha$, $\beta$ and $\gamma$ are the roots of equation $x^3-3x^2+3x+7=0$ ($\omega$ is the cube root of unity),then ...
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2answers
70 views

How to solve $(-3)^{1/3}$?

(z is a complex number )How to solve $(1-z)^3=-3$? at first I try to calculate $(-3)^{1/3}=?$ according the answer $(3)^(1/3)*e^(i*((pi) +2(pi)*k))$ but why? the radius is (3)^(1/3) the angle isn't ...
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1answer
29 views

Taylor expansion of $f(z)=\frac{z-1}{z^2-3z+3}$

We are given the function $f: \mathbb C \to \mathbb C$ defined by $f(z)=\frac{z-1}{z^2-3z+3}$ Is it possible to define $f$ as its taylor expansion near the point $z=i\sqrt 3$? If so, what is the ...
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1answer
48 views

Justify this equation?

$$\dfrac{e^{i\theta} -1}{e^{2i\theta} -1} = \dfrac{e^{\frac{i\theta}{2}} - e^{\frac{-i\theta}{2}}}{e^{\frac{3i\theta}{2}}-e^{\frac{-i\theta}{2}}} $$ However: $$e^{\frac{i\theta}{2} - ...
0
votes
1answer
24 views

Max value of a expression

given $M\subset\mathbb{R}$ where $M=\{|z^2+az-1|:z\in\mathbb{C}\wedge|z|=1\},a\in\mathbb{R}$ find max value of $M$ in function of $a$ i tried to make $z=e^{\theta i}$ $$\begin{align} ...
96
votes
11answers
6k views

Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...
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2answers
114 views

Help with Complex Number (De Moivre's theorem)

Please help with this question. "Use De Moivre's theorem to solve $z^5 = -1$. By grouping the roots in complex conjugate pairs, show that: $z^5+1 = (z+1)(z^2 - 2z\cos(\pi/5) +1) = (z^2 - ...
3
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1answer
69 views

Analytic exponential function problem [closed]

Let $x>0.$ How do i show that $\sum_{n=1}^\infty \exp{(-xn^2z)}$ defines an analytic function on $\Re(z)>0?$
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0answers
115 views

Help with Complex Number problem (Argand Diagram)

Can anyone help me with this question? "Use De Moivre's theorem to solve the equation $z^5 = 1.$ Show that the points representing the five roots of this equation on an Argand diagram form the ...
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3answers
111 views

Multiplication of a complex function with essential singularity with another complex function with a pole at the same point

Im trying to proof or disprove the following claim: If $f(z)$ and $g(z)$ are holomorphic in an annulus $0 < |z − z(\beta)| < R$ and $f$ has an essential singularity at $z(\beta)$ and $g$ has a ...
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0answers
30 views

Isometries of $\mathbb{R}^3$ by complex numbers.

I know that the isometries of $\mathbb{R}^3$ which preserve orientation and which fix one point, say origin, can be represented by complex numbers; they are described by the well known Mobius ...
2
votes
1answer
190 views

identity of $(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+…+(I-w^{n-1}zT)^{-1}]$

I am trying to understand the identity $$(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+...+(I-w^{n-1}zT)^{-1}] \quad (*),$$ where $T \in \mathbb{C}^{n\times n},z\in \mathbb{C}$ and the ...
2
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0answers
64 views

How to use complex numbers in geometry

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and ...
2
votes
2answers
56 views

Find and determine the type of all singularities of function $f(z)=\frac{z-1}{\sin(\frac{\pi}{z})\sin(\pi z)}$

Find and determine the type of all singularities of function $$f(z)=\frac{z-1}{\sin(\frac{\pi}{z})\sin(\pi z)}$$ I found that the set of the singularities is following: ...
1
vote
3answers
66 views

Is it possible to prove simply by manipulation that $\lim\limits_{x\to\infty} x^2\bigl(2-e^{-i/x}-e^{i/x}\bigr)=1$?

In fact, it turns out that $$\lim_{x\to\infty} x^2\left(2-e^{-z/x}-e^{z/x}\right)=-z^2$$ for any complex $z$. What is a simple (no DLH, Taylor, Maclaurin, ...) and "primitive" (no using directly or ...
4
votes
1answer
118 views

Complex integration $\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$

I'm trying to evaluate the integral $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$$ using complex numbers. Meaning, instead of calculating $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt,$$ I want ...
1
vote
2answers
176 views

De Moivre's theorem application

Suppose I am given a complex number $$z=\frac{(1+i\cos(x)+\sin(x))^n} {(1-i\cos(x)+\sin(x))^n}$$ and I need it to convert to simpler form First of all I know that a complex number is of form ...
2
votes
3answers
100 views

When the arguments of two roots of a quadratic equation are equal?

Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$ How can I obtain the condition for $$\arg z_1=\arg z_2$$ containing $a,b,c?$ At present I have, ...
19
votes
3answers
2k views

Sine of a Complex Number

While I know that $\sin(x)=2$ has no real solution, I tried seeing if it has a complex solution. That equality is equal to $$e^{2ix}-4ie^{ix}-1=0$$ Taking a quadratic in $e^{ix}$ I got ...
5
votes
1answer
97 views

Solve complex equation with exponential

I have to solve: $$e^{3z}+3ie^{2z}-ie^z+3=0$$ My attempt: Let $0\ne x:=e^z$. Then we can rewrite our equation as: $$x^3+3ix^2-ix+3=0$$ $$ix^2(-ix+3)+(-ix+3)=0$$ $$(-ix+3)(ix^2+1)=0$$ So $x\in ...
6
votes
2answers
97 views

Minimum of $f(z) = \left|z^2+z+1\right|+\left|z^2-z+1\right|$

For $z\in\mathbb{C}$, calculate the minimum value of $$ f(z) = \left|z^2+z+1\right|+\left|z^2-z+1\right| $$ My Attempt: Let $z= x+iy$. Then $$ \begin{align} z^2+z+1 &= ...
0
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1answer
77 views

Center of gravity of a regular polygon

How do I prove that the origin is the centroid of the regular polygon whose vertices are the solutions of the equation $z^n=1$ in the complex plane?
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3answers
70 views

Real and imaginary part of $ (1-i\sqrt{3})^6$

i am a bit stuck here. As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$ I already found out that the ...
3
votes
1answer
97 views

Picard's theorem applied to $f^n + g^n =1$

So I have the following problem. Part 1 is just to state Picard's theorem, so for that we have that any entire holomorphic function takes on every value with possibly one exception. Part 2 is to show ...
3
votes
3answers
64 views

Prove that $\frac{e^{2x}-1}{e^{2x}+1}i=\tan{ix}$

I have a doubt in complex numbers which I am unable to solve. The question is Prove that $$\left(\frac{e^{2x}-1}{e^{2x}+1}\right)i=\tan{ix}$$ I tried using hyperbolic sin and cosines but failed. Can ...
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1answer
70 views

Two different matrix representations of complex numbers

There are two different ways to represent a complex number with $2 \times 2$ real matrices: $$ \rho: \mathbb{C} \rightarrow M_2(\mathbb{R}) \qquad \rho(z)=\rho(a+ib)= \left[ \begin{array}{ccccc} ...
3
votes
2answers
102 views

Property of complex numbers.

Let $z \in \mathbb{C}$ such that $Re(z^{n})\geq0, \forall n\in\mathbb{N}$, where $Re(z^{n})$ is the real part of $z^{n}$. Show that $z\in\mathbb{R}^{+}$. If $z=a+bi$, $a,b\in\mathbb{R}$, then for ...
1
vote
1answer
310 views

What is this representation of complex numbers?

I was doing numericals on synchronous generators and came across this step in one of the examples. I have no idea what kind of math is used here. Can someone help? $(1.5 + 2.0j)\Omega = 2.5 \angle ...
0
votes
1answer
74 views

Cauchy Principal Value Integral calculation

How can i resolve this integral in Cauchy principal value? $$\int_{-\infty}^ \infty \! \frac{x+\sin x}{x(x^2+4j-4)^2} \ \mathrm{d}x $$ Then $$\int_{-\infty}^ \infty \! \frac{1}{(x^2+4j-4)^2} \ ...
1
vote
0answers
28 views

How to rewrite $PP^*$ in terms of sums of its column vectors?

$\begin{align} I & = & PP^* \\ & = & \begin{pmatrix} \bf{x_1} & \bf{x_2} &\dots & \bf{x_n} \end{pmatrix} \begin{pmatrix} \bf{x_1^*} \\ \bf{x_2^*} \\ \vdots \\ ...