Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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0
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3answers
58 views

$x=e^\frac{2i\pi}{11}$, show that $i\tan\frac{3\pi}{11}=\frac{x^3-1}{x^3+1}$.

$x=e^\frac{2i\pi}{11}$, show that $i\tan\frac{3\pi}{11}=\frac{x^3-1}{x^3+1}$. I don't know how the solution jump to this. Please help. Thank you.
0
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2answers
138 views

How to differentiate e to a function?

I am trying to find the derivative of $$\large e^{2 \pi i t \sin(\pi/(2t))}.$$ I know that I am to take the derivative of the exponent, and then multiply it by the beginning problem - the piece that ...
3
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1answer
213 views

Linear relations satisfied by nth root of unity

How do you characterize all the linear relations satisfied by $n$th roots of unity with real, integral and non-negative integral coefficients? Here are two examples for 3rd and 4th root: Let ...
0
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2answers
77 views

A question about a complex variable function

My question is about the function $f(z)=e^{-z^2}$. Is it everywhere continous? Holomorphic? Thanks, Dan
2
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2answers
154 views

Polar form of Complex numbers

I'm doing some work with complex numbers and I've come across this exercise in the "Polar form" section. $$(1/2+i(\sqrt{3}/2))^{100}$$ Of course this exercise is manageable with the help of Pascal's ...
1
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1answer
87 views

question about solution of equation complex variable

A friend just told me that the equation $e^{z^2}=0$ has solution. Is it true? Thanks, Dan
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12answers
3k views

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
1
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2answers
209 views

Proving that this equation represents a circle.

I'm trying to prove that the set of $z$ that satisfies the following equation represents a circle or a straight line on $\mathbb C$. $$(a\bar c-c\bar a)|w^2|+(a\bar d-c\bar b)w+(b\bar c-d\bar a)\bar ...
4
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2answers
117 views

Complex numbers: With conjugate

I've just started calculating complex numbers (last time I calculated with complex numbers was in high school) and I've already got stuck at this exercise: $$3z-i\bar z = 7-5i$$ where $\bar z$ ...
1
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1answer
78 views

Let $f(z)=e^x + ie^{2y}$ where $z=x+iy$. Where does $f'(z)$ exist?

Let $f(z)=e^x + ie^{2y}$ where z=x+iy is a complex variable defined in the whole complex plane. a)Where does f'(z) exist? b) Where is f(z) analytic? Answer: a) I used the Cauchy Riemann to test ...
1
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2answers
509 views

Complex solutions of $\sin z = i \alpha \cos z$

I'm trying to solve the following question: Let $\alpha$ $\in [-1, 1]$ be a real number. Find all complex numbers $z$ that satisfy the equation: $\sin z = i \alpha \cos z$ This is what I've done ...
1
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2answers
78 views

Multiplying imagionary roots of a polynomial

I am trying to answer the following question: The roots of the quadratic equation $ax^2-16x+25$ are $2+mi$ and $2-mi$, where $m>0$. Compute the sum of $a+m$. Should the zeros of the equation ...
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1answer
133 views

Prove Complex Function Is Holomorphic

Prove for $a\gt0$ that following series is holomorphic $$ \sum_{n=1}^\infty \frac {1}{(a+n)^z} \quad \textrm{for} \quad \operatorname{Re}z \gt 1 $$ I'm trying to prove this given that $Re \quad z ...
2
votes
2answers
72 views

complex numbers - proof of this statement

i am trying to prove this statement, i dont but how to start. $$\forall z,w \in \mathbb{C}\quad |z|^2+|w|^2=\frac{1}{2}(|z+w|^2+|z-w|^2)$$ can someone please show me how start?
0
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1answer
39 views

what is the difference - sorry for over-simplicity

i am asking too simple question, sorry for that. what is the difference between these two imaginär numbers? $\operatorname{Im}(| \sqrt2+3i|^2)$ vs. $\operatorname{Im}((\sqrt2+3i)^2)$ $| ...
2
votes
1answer
76 views

Evaluate a certain derivative

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let $$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
1
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2answers
104 views

Complex numbers - finding a square root of something

Let $z_1 , z_2 $ be two complex numbers that satisfy: $\dfrac{z_2 } {\bar{z_1}}= \frac{3}{8} \big(\cos(75^{\circ})+i\sin(75^{\circ})\big) $ , $z_1 z_2 ^2 = \frac{1}{3} \big(\cos(120^{\circ}) + ...
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2answers
56 views

Equation with Complex Numbers - Help!

Can someone help me solve the following equation? $$ 3z^3 + 2z^2 = 6z-4 $$ Thanks in advance!
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2answers
185 views

Finding a basis for a complex lattice given a nondivisible vector in the lattice

If I am given some lattice defined as, say $$L=\{Az_1+Bz_2\ |\ A,B \in\mathbb{Z}\}$$ and a vector $v=az_1+bz_2$ , where $\gcd(a,b)=1$, I would like to find another vector $\,w\in L\,$ such that ...
1
vote
1answer
111 views

exercise on complex numbers

Let $$f(z)=\frac{z-a}{z-b}$$ with $a,b\in D(0,r)$ and $r>0$. Let $$E=\{z\in\mathbb C: f(z)\notin N\}$$ $$N=\{Re(z)\leq 0;Im(z)=0\}$$ How can i find $E$ in terms of $r$?
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1answer
85 views

Finding a basis

Finding the basis for the kernel of: \begin{pmatrix} a & b \\c & d\end{pmatrix} $which$ $maps$ $to:$ \begin{pmatrix} a \\a\\3a + b \end{pmatrix} It's all complex, but I'm not sure if ...
1
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1answer
77 views

Question about derivatives of complex-valued functions

For $ z \in \mathbb{C}, t \in \mathbb{R}, \\f : \mathbb{C} \times \mathbb{R} \to \mathbb{C}, \\a : \mathbb{C} \times \mathbb{R} \to \mathbb{R}, \\b : \mathbb{C} \times \mathbb{R} \to \mathbb{R}$ And ...
1
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1answer
254 views

Complex Analysis and Limit point help

So S is a complex sequence (an from n=1 to infinity) has limit points which form a set E of limit points. How do I prove that every limit point of E are also members of the set E. I think epsilons ...
2
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1answer
85 views

Simple question about derivatives of complex functions

Is it true that for any complex function $f(z, t)$, the following equation is correct? $\frac{\partial f(z,t)}{\partial t} = \frac{\partial f^R(z,t)}{\partial t} + i \frac{\partial f^I(z,t)}{\partial ...
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4answers
2k views

Do the real numbers and the complex numbers have the same cardinality?

So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid. Can the approach be extended to say that the set of complex numbers has ...
3
votes
2answers
275 views

Require brilliant resources to self teach.

I'm far from the level of mathematical knowledge every user on this website posseses, however I am very much determined to get there as my love for mathematics increases. These are the topics: ...
2
votes
4answers
720 views

How are complex numbers useful to real number mathematics?

Suppose I have only real number problems, where I need to find solutions. By what means could knowledge about complex numbers be useful? Of course, the obviously applications are: contour ...
1
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1answer
93 views

Complex numbers- true of false?

If f(z) is an entire function, which gets only real values for real z, and $$ f(0)=0,$$ $$f'(0)\ne 0$$ and the Image of the imagainary axie is a straight line, then this line is the imagainary axie. ...
1
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1answer
153 views

Complex injective function

I'm trying to see if the function: $$z \mapsto z^n+\exp(ia) \cdot nz$$ is an injective function at the open unit circle. Please help.
7
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2answers
990 views

Limit of complex function

Im trying to find the limit of: $$ \frac{\operatorname{Re}(z) \operatorname{Im}(z)}{z^2}$$ as z tends to zero.
2
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3answers
83 views

Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root

Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root a with $|a| > 10$. how can i show that above statement is true/false.can ...
5
votes
4answers
166 views

Can a cubic that crosses the x axis at three points have imaginary roots?

I have a cubic polynomial, $x^3-12x+2$ and when I try to find it's roots by hand, I get two complex roots and one real one. Same, if I use Mathematica. But, when I plot the graph, it crosses the ...
0
votes
0answers
186 views

Modulus of a complex function: $\Psi(x)=A_0 e^{-kx^2} e^{i\alpha x}$

I am given $$\Psi(x)=A_0 e^{-kx^2} e^{i\alpha x}$$ Here, $\large A_0=[\frac{1}{\pi \sigma_0^2}]^\frac{1}{4}$, $\large k=\frac{1}{2\sigma_0^2}$, and $\large \alpha=\frac{p_0}{\hbar}$ I want to find ...
1
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1answer
42 views

Unwanted $i$ floating around when trying to calculate $\langle p\rangle$

$\def\sp#1{\left\langle#1\right\rangle}$I am given $$ \Psi(x,0)=A_0 \exp\left(-\frac{x^2}{2\sigma_0^2}\right) \cdot \exp\left(\frac{i}{\hbar}p_0x\right)\tag1$$ where $A_0=(\pi ...
1
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1answer
120 views

Simplified way of finding a complex number raised to another complex number

This question here has the answer but I'm still in school and I don't understand any of it. I'm writing a computer program that takes a complex number a + ib and raises it to c + id and I need to ...
2
votes
3answers
697 views

Find all complex numbers $z$ satisfying the equation

I need some help on this question. How do I approach this question? Find all complex numbers $z$ satisfying the equation $$ (2z - 1)^4 = -16. $$ Should I remove the power of $4$ of $(2z-1)$ and ...
3
votes
2answers
56 views

Curve of Equal SWR

I'm trying to figure out how radio frequency "matching stubs" work. In order to fully understand the problem, I need to know how the "curve of equal SWR" looks like. I did a few plots, and it looks ...
0
votes
2answers
356 views

Contour integration that is reduced to integration over unit circle

I want to evaluate $\displaystyle \frac{1}{2\pi}\int_{0}^{2\pi} \frac{1}{1-2rcos\theta + r^2} d\theta$ for $0 < r< 1$. I was thinking or replacing $2cos\theta = (e^{i\theta} + e^{-i\theta}) $ ...
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3answers
177 views

Non-existence of a certain holomorphic function on the unit disk

I am trying to prove the following: Let $n\in \mathbb{N}$. Prove that $\not \exists$ a holomorphic function $f$ on the open unit disk satisfying: $f\left(\displaystyle \frac{1}{n}\right) = 2^{-n}$ ...
1
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1answer
210 views

Isolated singularities, poles and removable singularities

Let $f$ be holomorphic with an isolated singularity at $z_0$. Suppose that $\exists M,m,\epsilon$ positive numbers such that $|f(z)| \leq M|z-z_0|^{-m}$ for $0<|z-z_0|<\epsilon$. Prove that ...
4
votes
3answers
145 views

Complex series: $\sum_{n=0}^\infty\left( z^{n-2}/5^{n+1}\right)$ for $0 < |z| < 5$

How would one compute $$ \sum_{n=0}^\infty\frac{z^{n-2}}{5^{n+1}} $$ where $0\lt|z|\lt5$? I have literally no idea where to start, all I know is that the answer will not have summations. Any help ...
22
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2answers
736 views

What do we lose passing from the reals to the complex numbers?

As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
2
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1answer
174 views

In solving $n^{\text{th}}$ roots of unity, for $e ^{2ki\pi}$ , $\;k = 0,\pm1, \pm2, \ldots$ , why must “k” must be an integer?

In solving $n^{\text{th}}$ roots of unity, when using the expression $e ^{2ki\pi}$ , $\;k = 0,\pm1, \pm2, \ldots$ , why must "$k$" must be an integer? I understand there's a substitution going on, ...
1
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0answers
197 views

The general recipe for finding the conjugate of a complex function

I have the general recipe for finding the complex conjugate of a function down as follows: Suppose I have $f(z)$: Separate $f(z)$ into a sum of real and imaginary functions such that ...
2
votes
1answer
96 views

Exponentials and linear operators

I don't quite understand how the following works, could anyone please explain? Let $O_i,i=1,2,3$ be linear operators on a space of functions. They satisfy the commutation relation ...
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1answer
39 views

proving that $|e^z|$ is smaller than $1$ if $\text{Re}(z)$ is nonpositive

Question : Given $\text{Re}(z) \le 0$ prove that $|e^z| \le 1$. Try: $z=x+yi$, it's given that $x \le 0$. $$|e^{z}| = |e^{x+yi}|=|e^xe^{yi}|=e^x|e^{yi}|,$$ with $e^x \le e^0$ because $f(x)=e^x $ is ...
1
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2answers
272 views

Cauchy-Schwarz Inequality for Complex Numbers

Simple question: do we really need the conjugate in the inequality? $$ |\sum_{j=1}^n a_j \overline{b_j}|^2 \leq \sum_{j=1}^n |a_j|^2 \sum_{j=1}^n |b_j|^2 $$
1
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2answers
61 views

Polar coordinates - issue with direction denoted by angle

Convert $1-\sqrt{3}i$ to polar coordinates $(r,\varphi)$. I started by computing $r=|1-\sqrt{3}i|=\sqrt{1^2+\sqrt{3}^2}=\sqrt{4}=2$. When I tried to compute the angle I did something like ...
2
votes
4answers
146 views

Can the series $1/(i-0) + 1/(i-1) + 1/(i-2) + \cdots$ be reduced to $\log(i)$

Can the series $\dfrac1{i-0} + \dfrac1{i-1} + \dfrac1{i-2} + \cdots$ be reduced to $\log(i)$? It looked similar to the harmonic series, so I checked wikipedia for Harmonic series, and found the ...
1
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3answers
99 views

Proof of a complex identity

This might be a very obvious one, but I am stuck on this from a long time. If $F(s) = M(s) + N(s)$ where $M(s)$ is even polynomial function and $N(s)$ is odd polynomial function (where $s$ is a ...