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
109 views

Why is Euler's formula valid for all $n$ but not De Moivre's formula?

The Wikipedia page on De Moivre's Formula says the formula doesn't hold for non-integer $n$, since non-integer powers of a complex number can have multiple values. It then goes on to say that this ...
0
votes
1answer
65 views

How can I prove this equation?

My prof put this equation on the board, without any kind of explanation or proof. When I asked him for one, he didn't really give me a solid answer. $$w = r(\cos \theta + i\sin \theta)$$ Then $w^n ...
0
votes
0answers
29 views

Guidance for complex numbers/analysis problem needed [duplicate]

I'm looking at this one problem in a book of mine, but I can't even seem to start it. Let $z_1,z_2,...$ be a countable set of distinct complex numbers. If $|z_j-z_k|$ is an integer for every ...
1
vote
0answers
56 views

Complex roots of equation $z^\mu=r$

Find the complex roots of equation concerning unknown complex number $z$ \begin{equation} z^\mu=r,\quad \mu>0,r\in\pmb{R} \end{equation} A solution given by a book is to only consider the ...
0
votes
3answers
35 views

Question regarding in periodic function

I have question I know that $\cos(x+2\pi)=\cos x$ and $\sin(x+2\pi)=\sin x$ but if we have $\cos(x+\pi)=?$ and $\sin(x+\pi)=?$ with explaination thanks
2
votes
2answers
122 views

Solve cos(z) + sin(z) = i, where z is a complex number and i the imaginary unit

So yeah everything is in the title, I tried the trigonometric identity with sin(a+bi) and cos(a+bi) and I tried changing sin(z) and cos(z) for their complex expression, but all to no avail EDIT: ...
3
votes
3answers
182 views

Proving $\arg(zw)=\arg(z)+\arg(w)$

This is my attempt I know this is incomplete or may even be wrong. Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient ...
0
votes
2answers
48 views

Finding argument to complex number?

I'm reading a bit on complex numbers, but haven't deal with trigonometry a lot before, so here's my question; how do I calculate the argument of a complex number when the sin and cos of the argument ...
6
votes
6answers
168 views

What's the result? $1/i=?$, where $i=\sqrt{-1}$ [duplicate]

I just had my first math class in the university, and I understood everything pretty well, but I think I have misread this one because I read that the result is $-1$. Thanks for your answers!
0
votes
0answers
128 views

Given roots on the unit circle, find the complex reciprocal polynomial

Given "all" the $m$ zeros on the unit circle of a complex reciprocal polynomial of even degree $2N > m$, can we find the polynomial? The known conditions are: We have all the $m$ zeros on the ...
2
votes
1answer
60 views

Find loci of the points in complex plane such that $\mathrm{Im}(\frac{z-z_1}{z-z_2})^n=0$

Find loci of the points in complex plane such that $$\mathrm{Im} \left (\frac{z-z_1}{z-z_2}\right )^n=0,$$ where $n\in\mathbb{N}$, $z_1, z_2$ are the given points in $\mathbb{C}$. When $n=1$, it ...
2
votes
2answers
46 views

Proving $\mathbb Z[i]$ is euclidean domain .

From the definition of euclidean domain , one has to select euclidean function . Let $\mathbb Z[i]=\{a+bi | a,b\in \mathbb Z,i=\sqrt{-1}\}$ We have to select an euclidean function $f$ , such that ...
0
votes
2answers
140 views

how to find inverse point in a complex plane

How to find the inverse point of the point z=a with respect to the circle $|z-c|=r$ (where c is the origin and r the radius) ? $c+\frac{r^{2}}{a-c}$ this is the answer given in the book...how do we ...
2
votes
1answer
64 views

Proof of equality of complex numbers needed?

Is $a+bi=c+di\iff a=c, b=d$, where $a,b\in\mathbb{R}$ something that requires proof? My instinct is telling me that proof is required to demonstrate that playing around with real numbers (using field ...
2
votes
3answers
56 views

Roots of Unity - $x^3 = -i$

I need to find the roots of unity for the following equation: $$x^3 + i = 0$$ Thus, $x^3 = -i$. I know that $-i = \exp[i(\frac{3\pi}{2} + 2n \pi)]$ however I do not know how to get all roots. ...
1
vote
3answers
275 views

Every line or circle in $\mathbb{C}$ are solution sets to the equation…

Here is a complex analysis homework problem I can't quite figure out: Prove that every line or circle in $\mathbb{C}$ is the solution set of an equation of the form $a|z|^2+\bar{w}z+w\bar{z}+b=0$, ...
0
votes
1answer
27 views

Sub-space problem in complex space

A problem in my professor's guide is driving me nuts, I don't even know where to start, this is the problem: Is $ \{(z,u) \in \mathbb{C}^2 / z - \overline{z} + u = 0\} $ a sub-space of $ ...
4
votes
3answers
2k views

Find all values for cos(i)

In my Differential Equations class recently we have learned about Euler's Formula and Fourier Series. I am given the problem ...
0
votes
2answers
37 views

Complex roots of equation with geometric sequence?

1 + z + z^2 + z^3 = 0 How to find the solutions to this one? My workbook tells me to use the fact that it's a geometric sequence, but I haven't worked with them at all, so I am not sure I know what ...
0
votes
0answers
56 views

complex limits, how to show they go to 0?

In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are: $e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number ...
2
votes
2answers
45 views

Complex analysis: Rewrite $\cos^{-1}{i}$ in algebraic form

I'm stuck in this problem (complex analysis), my answer is not the one reported in the book: Rewrite $\cos^{-1}{i}$ in the algebraic form. A: $k\pi + i \frac{\ln{2}}{2}\ \forall\ k \in \mathbb{Z}$ ...
0
votes
1answer
78 views

Complex function representing rotation of the plane

The function $f(z) = \dfrac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. I'm not even sure what this question means. I need help.
1
vote
4answers
51 views

Third point of a triangle in the complex plane

I have an equilateral triangle with two points equal to $(2+2i)$ and $(5+i)$. I want to find the third point(s) (there are $2$ of these). I have that the side length of the triangle is $\sqrt{10}$.
-2
votes
1answer
62 views

Complex numbers and geometry

There exist two different complex numbers $c_1$ and $c_2$, that together with $2+2i, 5+i$ form the vertices of two equilateral triangles. Find the product $c_1c_2$.
0
votes
1answer
71 views

How to express the equation of the line joining the complex numbers $-5 + 4i$ and $7 + 2i$ n terms of $z$ and $\bar z$?

The equation of the line joining the complex numbers $-5 + 4i$ and $7 + 2i$ can be expressed in the form $az + b \overline{z} = 38$ for some complex numbers $a$ and $b$. Find the product $ab$. I ...
1
vote
1answer
36 views

how should I think about the complex-exponential form of sinusoid waves?

Say there's a sinusoid wave with amplitude $A$, frequency $\omega$, and phase shift $\psi$, then one way to write it is $A cos(\omega t - \psi)$. But it can also be written as $Re(Ae^{i(\omega t - ...
0
votes
3answers
77 views

Complex Number Geometry

Problem: There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $2+2i$, $5+i$, and $c$ form the vertices of an equilateral triangle. Find the product $c_1c_2$. I've been struggling with ...
0
votes
1answer
29 views

Calculating Bloch-Wigner dilogarithm

Is there some tool/calculator (or some tables) for explicitly calculating values of the Bloch-Wigner dilogarithm?
1
vote
1answer
31 views

Absolute values and inequalities

So I've been trying to solve this one for a few hours and am now out of ideas on how to approach this problem. Here are the inequalities: $$\text{show that if}$$ $$z,w \in \Bbb C$$ $$|z| < ...
2
votes
1answer
45 views

Why $\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 \in \mathbb R \iff z_1, z_2, z_3 \text{ are along the same line}$

Prove: $$\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 \in \mathbb R \iff z_1, z_2, z_3 \text{ are along the same line}$$ My attempt: Since $z+\bar{z} =2 \operatorname{Re}z \in \mathbb R$, so we can ...
1
vote
3answers
125 views

Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+…+\cos (n-1)x$ [duplicate]

By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ...
1
vote
2answers
84 views

Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?

Is there anything special with the form: $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers. In fact, I saw it in a problem: If $|z| = 1$, prove that ...
0
votes
2answers
35 views

Simplify $w=\frac{(1+i)z-i+1}{iz-1}$

I have difficulties understanding how this expression $$w=\frac{(1+i)z-i+1}{iz-1}$$ is simplified to this $$w=1-i-2\cdot\frac{1+i}{z+i}$$ Here are some steps from my exercise notebook: ...
0
votes
0answers
31 views

Behavior of lower incomplete gamma function at complex infinity

The lower incomplete gamma function is given by $ \gamma\left(s, x\right) = \int\limits_0^x t^{s-1} e^{-t} {\rm d} t~,$ and has a well-defined analytic continuation for both $s$ and $x$ [1]. ...
1
vote
2answers
126 views

Solving complex equation for z?

How do you solve equations involving $z = a + bi$ and imaginary units? The one I am looking at right now: $$\frac{z-2}{z+1} = 3i$$ If you could help me with this one, I think I can do the rest by ...
0
votes
0answers
98 views

Smart way to Integrate $\int \dfrac{dx}{1+x^4}$ [duplicate]

I know abt that straightforward lengthy way. anyone knows a smarter way
0
votes
1answer
54 views

On complex numbers and absolute values

Exercise 1.31 of Analysis by Apostol states: Given three complex numbers $z_1,z_2,z_3$ such that $|z_1| = |z_2| = |z_3| = 1$ and $z_1 + z_2 +z_3 = 0$. Show that these numbers are vertices of an ...
1
vote
4answers
87 views

Show that if $z,w\in\mathbf{C}$, $|z|<1$ and $|w|<1$, then $\left|\frac{z-w}{1-\overline{z}w}\right|$<1? [duplicate]

Every way I try to approach this turn it into proving the inequality $|z-w|<|1-\overline{z}w|$. Not sure at all how to approach it at the moment.
0
votes
0answers
90 views

Can this expression be evaluated for odd $n>1$?

I suppose $n$ is an odd natural number greater $1$ and e.g. squarefree. Now I try to evaluate/simplify this expression where $\mu(m)$ is the Möbius function : $$\prod_{d|n,\ ...
0
votes
1answer
42 views

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$ I have concluded that $xu_x = yu_y$. Not sure how to proceed ...
6
votes
6answers
260 views

Solve $\cos{z}+\sin{z}=2$

I am trying to solve the question: $\cos{z}+\sin{z}=2$ Where $z \in \mathbb{C}$ I think I know how to solve $\cos{z}+\sin{z}=-1$: $1+2\cos^2{\frac{z}{2}}-1+2\sin ...
0
votes
0answers
35 views

$√2|z|≥|Rez|+|Imz|$ [duplicate]

I just came across this simple inequality which I am finding it difficult to prove. For any complex number z I need to prove that the following inequality holds $$√2|z|≥|Rez|+|Imz|$$ I would much ...
3
votes
1answer
72 views

Suppose that $a_0 >a_1 >…>a_{2013} >0.$ Prove that $\sum_{n = 0}^{2013}a_nz^n \neq 0$ when $|z|<1$ [duplicate]

Suppose that $a_0 >a_1 >...>a_{2013} >0.$ Prove that $\sum_{n = 0}^{2013}a_nz^n \neq 0$ when $|z|<1$ Not sure where to begin with this. Any suggestions? Thanks.
1
vote
2answers
47 views

Finding $\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$

$\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$ I am having trouble getting a final answer that makes sense to me. Here is what I tried: ...
1
vote
0answers
47 views

Prove that every complex number is in the range of the entire function $e^{3z} + e^{2z}.$

Prove that every complex number is in the range of the entire function $e^{3z} + e^{2z}.$ By Picard we have that every number except maybe one is, but that is all I've got. Help would be great! ...
0
votes
2answers
38 views

How to prove property of complex powers

I have the following problem: If $b$ is real, prove that $|a^b|=|a|^b$. In this case, $a$ is complex number. I know the definition of a complex power, $a^b=e^{b\log(a)} $, but I´m not sure how to ...
0
votes
1answer
21 views

Sketching points given by complex numbers

I cannot remember much about circles. If I have $|z-1+i|=1$, how do I translate this geometrically. I know it's a circle but I can't remember how to do this.
1
vote
2answers
38 views

Sum of two trig function's identity

We all know that $\sin(x) + \sin(y) = 2\sin((x+y)/2)\cos((x-y)/2)$ But is there an identity for $\sin(x) + z\sin(y) = ?$ Or do I need to figure it out using Euler's formula $\sin(x) = (e^{ix} - ...
2
votes
2answers
122 views

Verify that $\sqrt{2}\left\| z \right\| \ge \left|\Re(z)\right| + \left|\Im(z)\right|$

Verify that $\sqrt{2}\left\| z \right\| \ge \left|\Re(z)\right| + \left|\Im(z)\right|.$ I started off noting that $z=x+iy$ and that $Re(z)=x$ and $Im(z)=y$ Then I know that I have to square both ...
6
votes
4answers
200 views

Can I conjugate a complex number : $\sqrt{a+ib}$?

Can I conjugate a complex number: $\sqrt{a+ib}$ ? Actually my maths school teacher says and argues with each and every student that we can't conjugate $\sqrt{a+ib}$ to $\sqrt{a-ib}$ because ...