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Questions tagged [complex-numbers]

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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2answers
41 views

How can we define the set of complex numbers without first defining multiplication and addition on the complex numbers?

Defining the Complex Numbers Usually, we are required to fully define a set before we can define a relation on that set. It would be very strange to define a relation in an as yet undefined domain, ...
2
votes
2answers
24 views

Complex Integral $\int_{C} \frac{e^{iz}}{z^3} dz$ using Cauchy Integral Formula of Derivatives

I am trying to find $\int_{C} \frac{e^{iz}}{z^3} dz$ on circle of $|z| = 2$ traversing once on positive direction. My approach was using Cauchy derivative formula $f^{(n)}(a) = \frac{n!}{2\pi i} \...
1
vote
0answers
15 views

Differential Equations: How to categorize graph and clockwise vs. counter-clockwise from eigenvalues?

I'm studying for my Final and having a hard time understanding the criteria for category (Sink, Spiral Sink, Center, etc.) and how to tell whether the direction is clockwise (CW) or counter-clockwise (...
0
votes
1answer
13 views

Convert the function into series

I have the function $$ f(z)={\frac{z}{z^2-2z+5}}, z_0=1 $$ i.e. I need to have it in form $f(z)={\sum{C_k(z-z_0)^n}}={\sum{C_k(z-1)^n}}$ I do $$ z-1=t, z=t+1, $$ $$ {\phi}(t)=f(t+1)={\frac{t+1}{(t+1)^...
1
vote
0answers
23 views

If the set is a complex linear space?

The set $P$ consists of complex trigonometric polynomials $$f_k(t)=\sum_{j=0}^k a_j e^{iτ_jt},$$ where $$ k\in \mathbb N, a_1,...,a_k \in \mathbb C, τ_1,...,τ_k \in \mathbb R.$$ Show that $P$ is a ...
1
vote
0answers
15 views

Questioning the commutative property in complex powers [duplicate]

We have:$$e^{2\pi\cdot i(1+\frac{1}{x})}$$ Power properties state that: $a^{b\cdot c}=(a^{b})^{c}=(a^{c})^{b}$. Thus we could re-write the above power as: $$(e^{2\pi\cdot i})^{1+\frac{1}{x}}=1$$ ...
-2
votes
0answers
25 views

How to solve for complex $z$? [on hold]

Which complex numbers satisfy the following equations? $|z|-(6+8i)z=-5.8579+140.000i$ $(6+9i)|z|-z=86.8375+107.2562i$
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votes
3answers
35 views

What's the difference between “real” and “imaginary” parts of a complex numer?

I was reading Go's documentation for the complex128 and complex64 data types when I came across something odd: "complex128 is the set of all complex numbers with float64 real and imaginary parts." ...
-2
votes
2answers
35 views

Value of $\displaystyle\sum_{n=1}^{39} \omega^{3n}$, where $\omega \neq 1$ and $\omega^5 =1$ [on hold]

Let $\omega$ be a complex number such that $\omega\neq 1$ and $\omega^5=1$. What is the value of $$\sum_{n=1}^{39} \omega^{3n}?$$ By the way I haven't learn anything about complex numbers before.
1
vote
2answers
30 views

Having a homogenous system of linear equations with real coefficients with a non-trivial complex solution than there is a real solution too.

I'm struggeling a bit with this proof. Suppose we have a homogenous system of linear equations with real coefficients with a non-trivial complex solution than there is a real solution too. This ...
0
votes
1answer
15 views

How to find the mapping of a region, complex analysis. [on hold]

Having a hard time solving this task: Find the image of the region $\text{{$z$ : Re($z$)>$0$}}$ under the mapping $w$=Ln$z$. I understand that I am supposed to write a function $f$ with ...
1
vote
3answers
72 views

What about the solutions of $z^{1/3} +1 = 0$?

I'm trying to find the zeros of the equation $$z^{1/3} +1 = 0.$$ My professor said that the solutions are the third roots of unity multiplied by $-1$. My problem is that when I calculate the cubic ...
0
votes
1answer
28 views

Complex Analysis on Holomorphic Anti-derivatives

Question: Let $U_{1} \subseteq U_2 \subseteq U_{3} \subseteq ... \subseteq \mathbb{C}$ be connected open sets and let $U = \cup_{i = 1}^{\infty} U_i$. Let $f$ be holomorphic on $U$. Suppose for each $...
2
votes
2answers
40 views

If $z=4+i \sqrt{7}$ then find the value of $z^3 -4z^2 -9z + 91$.

So i was learning complex numbers and i came across this problem. In the solution they have made $z-4=i\sqrt{7}$ and then they squared the above equation resulting in $z^2 -8z+16=-7$ then they ...
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votes
1answer
51 views

How do I solve this $z^5 + z^3 = iz^2 + i$

How can I find out what z is and the zero of this function? $z^5 + z^3 = iz^2 + i$ I tried everything but I cannot solve it... Thanks in advance!
0
votes
1answer
56 views

How to find all complex solutions?

When i'm solving this Equation $$z^{2018} + |z|^{1995}* \bar{z}^{83} = 2$$ I got this: $$z_{k} = (cos(\frac{2\pi k}{1995}) + isin(\frac{2\pi k}{1995}) * \frac{1}{\sqrt[2018]{cos(\frac{4202\pi k}{1995}...
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votes
1answer
33 views

Finding inverse to elements

I am trying to find the inverse of the following elements $2+\sqrt7$, $\sqrt3-\sqrt2$ and $1+i\sqrt5$. I would be very much thankful if someone could help me with this one.
1
vote
2answers
44 views

Convert $(1+i) ^ {1+i}$ to polar form

Can someone please help me understand the exponent/logarithm relationships to get through this problem? Thank you.
0
votes
2answers
36 views

Show $\overline{e^{i\theta}} = e^{-i\theta}$ using trig identities

My book claims we can verify this using "standard trigonometric identities." The notation of the right-hand side is throwing me off. Does that mean the complex conjugate of $e^{i\theta}$ equals its ...
3
votes
1answer
67 views

Prove that $\prod_{r=1}^m \sin \left( \frac {r\pi}{2m+1}\right) =\frac {\sqrt {2m+1}}{2^m}$

Prove that $$\prod_{r=1}^m \sin \left( \frac {r\pi}{2m+1}\right) =\frac {\sqrt {2m+1}}{2^m}$$ My try: $$\prod_{r=1}^m \sin \left( \frac {r\pi}{2m+1}\right) =\prod_{r=1}^m \left(\frac {e^{\frac {ir\...
0
votes
2answers
22 views

How to take an absolute value or modulus of z?

let's assume the : $$z=\frac{e^{(-jc)}}{(a+jb)}$$ I would like to take the absolute value of z. I started with multiplication z with $\frac{(a-jb)}{(a-jb)}$ and got: $$abs\frac{e^{(-jc)}}{(a+jb)}=\...
1
vote
2answers
29 views

Why is $\int_{-1}^1 (1-|t|) \cdot e^{-i \omega t} dt$ equal to $2 \cdot \int_0^1 (1-t) \cdot \cos{\omega t}\, dt$

I have a task where I should calculate the fourier transform of $$ \Delta(t) = \begin{cases} (1-|t|)& |t| \le 1 \\ 0 & |t| > 1 \end{cases} $$ The solution says, that $$ \int_{-1}^1 (1-|t|)...
1
vote
2answers
43 views

A line integral equals zero implies a real integral also is zero

I'm asked to check that the following line integral is zero: $$\int_{C(0,r)} \frac {\log(1+z)}z dz=0$$ (where $C(0,r)$ is the circle of radius $r$ centered at $0$) and then to conclude that for ...
0
votes
0answers
20 views

Series representation for a specific range

I am wondering if there is a valid series representation using: $f(z) = \sum_{k=-\infty}^{\infty} a_k(z-z_0)^k$ for $r<|z−z_0|<R$ Why is this not possible?
0
votes
1answer
28 views

Find $x$ and $y$ in terms of real and imaginary parts of $f(z)=\frac{1}{z}$

Find $x$ and $y$ in terms of real and imaginary parts of $f(z)=\frac{1}{z}$. Here is what I have tried so far: $$f(z)=\frac{1}{z}=\frac{1}{x+iy}=\frac{x-iy}{x^2+y^2}=u(x,y)+iv(x,y) \\ \implies u(x,y)=...
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votes
2answers
75 views

Complex number ,satisfies $|z-4/z|=3$ . What is the maximum value of $|z|$? [on hold]

Complex number, satisfies $|z-4/z|=3$ . What is the maximum value of $|z|$ ?
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votes
0answers
53 views

A strange proof of $\sqrt{-1}=1$ [duplicate]

I know that $\sqrt{-1}=i$,but I find a strange prove that it equals to 1. $$\sqrt{-1}=(-1)^\frac{1}{2}=(-1)^\frac{2}{4}=\sqrt[4]{(-1)^2}=\sqrt[4]{1}=1$$ Yes, it looks so strange and unreasonable. But ...
1
vote
0answers
63 views

Understanding a proof. Why is it necessary that $\alpha<1$?

I'm reading the following proof from Lang's book of complex analysis. But I don't know why is it necessary that $\alpha<1$?. I don't know where "$\alpha<1$" is used . I don't understand either ...
0
votes
2answers
38 views

$z \in \mathbb{C}^-$ where $\mathbb{C}^- := \{z \in \mathbb{C} : Re(z) < 0\}$ identity

Why is $|2-z|^2 = 4 + |z|^2 - 4Re(z)$, where $z \in \mathbb{C}^-$ and $\mathbb{C}^- := \{z \in \mathbb{C} : Re(z) < 0\}$. Why is this true?
1
vote
1answer
25 views

Help to show that $\lim_{|z| \to \infty} \exp(e^{\alpha|z|}- \varepsilon (e^{\beta \mathrm{Im}(z)}+e^{-\beta \mathrm{Im}(z)} ))=0$

I'm reading a proof of a theorem and I have problems with a limit in that proof. Let $0<\alpha < \beta < 1$ and $\varepsilon>0$. Can you help me to show that $$\lim_{|z| \to \infty} \exp(...
-1
votes
1answer
29 views

Sine of complex numbers.

It is stated that the system, whose displacement is defined by sin[√(A²-1) + X ], is at rest when A is greater than 0 and smaller than 1. How can this be shown?
0
votes
1answer
31 views

$|e^{z_1}-e^{z_2}|$ is less than $|z_1-z_2|$ if the real parts are non-positive

Is $|e^{z_1}-e^{z_2}| \le|z_1-z_2|$ if the real parts of $z_1,z_2$ are non-positive? I think yes, but what method do I proceed with? Do I proceed with the fact that modulus equals the number ...
-1
votes
0answers
64 views

Can the complex numbers be extended from an unsolvable equation in the complex domain? [closed]

In reference to the recently published pre-print https://arxiv.org/abs/1811.12175. The author proposes the extension of the complex numbers from an unsolvable equation in the complex domain, for ...
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votes
1answer
31 views

Complex number identity question

If $n$ is even and $z=\cos\theta+i\sin\theta$. By expressing $z^n$ in two ways, show that $$\binom{n}{0}-\binom{n}{2}+\binom{n}{4}-\cdots+(-1)^{\frac{n}{2}} \binom{n}{n}=2^{\frac{n}{2}}\cos\left(\...
0
votes
1answer
35 views

How does this expression in complex variable hold?

If $z_1, z_2, z_3$ are vertices of an equilateral triangle then $$|z_1-z_2|=|z_2-z_3|=|z_3-z_1|$$. But how does $\frac{1}{z_1-z_2}+\frac{1}{z_2-z_3}+\frac{1}{z_3-z_1}=0$ hold?
1
vote
0answers
17 views

Proving inequality relation between two complex numbers and a positive real parameter. [duplicate]

Question: Prove that: $$|z_1+z_2|^2 \le (1+c)|z_1|^2+(1+{1\over c})|z_2|^2$$ where $z_1,z_2$ are complex numbers and $c$ is a positive real parameter. Solution: We can write $$|z|^2=z\...
0
votes
1answer
23 views

Writing linear combination of exponentials as cosine

Show that $$y = A_1e^{ix} + A_2e^{-ix}$$ can be written as $$y = A\cos(x - \delta)$$ where A and $\delta$ are real. So far I have done the following: $$y = (A+B)\cos(x) + (A-B) i\sin(x)$$ Am I on ...
0
votes
1answer
33 views

irrational root of a number number [closed]

Let $x>0$, and let $\alpha$ be an irrational number. Can we make sense of $x^{\alpha}$ ? What about the case $x<0$ ?
3
votes
1answer
23 views

Congruence proof. Show $a$ is odd, $b$ is even, and $ a \equiv 1$ mod $4 $ under certain conditions.

If $ p \equiv$ $1$ mod $4$, $ p = a^2 + b^2$, and $a + bi \equiv 1$ mod $2+2i$, then $a$ is odd and $b$ is even. Moreover, if $4|b$, then $ a \equiv 1$ mod $4$, and if $ 4 \nmid b$, then $ a \equiv -1$...
0
votes
1answer
18 views

When is the conjugate of a function equal to the function of the conjugate of the argument ($f(x^*) = f^*(x)$)

Is there a special property that unites all functions for which $f(x^*) = f^*(x)$ holds? Naively I can show that it is true for any function that has a series expansion with only real coefficients, ...
0
votes
1answer
43 views

Reverting multiple rotations of an (irrational) angle

Problem Let $a\in\mathbb{C}$ , $|a|=1$ and $c\in\mathbb{R}$. Let us assume that $\exists{n\in\mathbb{N}} \;\; a=\exp(c i n)$ Find the value of smallest possible $n\in\mathbb{N}$. Background I ...
7
votes
2answers
373 views

Finding magnitude of a complex number

$$z = \dfrac{2+2i}{4-2i}$$ $$|z| = ? $$ My attempt: $$\dfrac{(2+2i)(4+2i)}{(4-2i)(4+2i)} = \dfrac{4+12i}{20} = \dfrac{4}{20}+\dfrac{12}{20}i = \dfrac{1}{5} + \dfrac{3}{5}i$$ Now taking its ...
1
vote
1answer
28 views

Is the determinant of a complex matrix the complex conjugate of the determinant of it's complex conjugate matrix?

Apologies for the confusing title. Suppose we have some square matrix $A$ with complex entries and it's conjugate matrix $\bar{A}$ whose entries are the complex conjugate of those in $A$. Is it ...
0
votes
0answers
24 views

Example of a Riemann mapping without continuous extension at the boundary.

Let $U$ be a simply connected open subset $U$ of $ \mathbb{C}$ and $D=\{z \in \mathbb{C}: |z|<1 \}$. I read a theorem which says that a Riemann mapping $f:U \to D$ can be continuously extended to ...
0
votes
2answers
36 views

How should I determine all numbers $z\in \mathbb C$ such that $z^3 = 4\overline z$?

Find all $z\in \mathbb C$ such that $z^3 = 4\overline z$ I have set $z = re^{i\theta}$ and found that $z^3 = 4(x-iy)$ leads to the equation $z = 2e^{-i\theta}$. I am supposed to determine all ...
3
votes
1answer
59 views

$\mathbb Z[\frac{2+i}{5}]\cap \mathbb Q =\mathbb Z$

As it says in the title I want to show that $\mathbb Z[\frac{2+i}{5}]\cap \mathbb Q =\mathbb Z$. Set $\omega=\frac{2+i}{5}$. $\mathbb Z[\omega]$ is the smallest ring that contains $\mathbb Z$ and $\...
1
vote
1answer
26 views

Is the principal root defined at the right half-plane continuous at $0$?

I'm reading a proof of a theorem. In the proof the author says that the principal root $z^{1- \delta/2}$ (that is, the function $z^{1- \delta/2}:=\exp(\mbox{Log}(z)(1- \delta/2))$, where $\mbox{Log}$ ...
6
votes
1answer
73 views

Why does calculating $\exp z$ using $\ln z$ via newton-raphson method fail to converge?

I am trying to calculate $\exp z$ using $\ln z$ via Newton-Raphson method $$x_{n+1} = x_n-\frac{f(x_n)}{f^{'}(x_n)}$$and got the formula $$x_{n+1}=x_n-\frac{\ln x_n-z}{\frac{1}{x_n}}$$ where $z = a + ...
0
votes
2answers
28 views

Proving the square root of $z\in \mathbb{C}$ geometrically

I hope somebody could help me to prove that the squareroot exists for all numbers in $\mathbb{C}$. A complex number is Always a stretch and a Rotation at the same time. For example (3,2) Projects (1,0)...
1
vote
2answers
29 views

How to factorize $zz^*-4z-4z^*+12=0$ (where $z^*$ is the complex conjugate of $z$)

I'm trying to factorize this: $$zz^*-4z-4z^*+12=0$$ to get this: $$|z-4|^2 - 4 = 0$$ where $z=x+yi$ is a complex number and $z^*=x-yi$ is the conjugate complex number of $z$. I'm trying to factorise ...