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

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Defining the Complex numbers

I posted this question nearly 10 days ago, but am still really not satisfied with the answers I got, I have no prior education in abstract algebra, group theory, or other abstractions, and most of the ...
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2answers
115 views

A question on complex numbers

We are given If $\cos(a+ib)$=$r (\cos\theta +i\sin\theta)$ then prove that $e^{2b} = \sin(a-\theta)/­\sin(a+\theta)$ I just tried and got $b = 0$ such that $\cos(a) = ra$. Will there be other ...
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2answers
707 views

Why isn't $\log(-1)=i\pi$?

Reading http://people.math.gatech.edu/~cain/winter99/ch3.pdf, $\log(z)$ is defined as $=\ln|z|+i\arg(z)$. Looking on the Wessel plane, isn't $\arg(-1)=\pi$ (more generally $\pi \pm 2 \pi n$)? And ...
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3answers
4k views

How to figure out the Argument of complex number?

I have the absolute value of complex number , $$ r = |z| = \sqrt{x^2 + y^2}$$ when $z = x + iy$ is a complex number. How can I calculate the Argument of $z$? Thanks.
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1answer
279 views

Decompose a complex symmetric matrix to retain positive definitness

I have a complex symmetric matrix $A$, (i.e. non-Hermitian and obeying $A=A^T$), which is positive definite, in the sense that: $$\Re({z^HAz}) > 0$$ for any $z$. I am able to verify this ...
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1answer
97 views

Math question complex numbers?

I have to find the roots of $(i)^{1/6}$ ...so I find $k= 0, 1, 2, 3, 4, 5$... the angle is zero degrees apparently...so the first root is $i^{1/6}\times [\cos (0+2\times 0\times \pi)/6 + i\times ...
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2answers
83 views

Math question complex number help?

Write the following numbers as an $(\alpha + \beta i)$ which means as an algebraic expression : $[2(\cos5 + i\sin5)]^{12}$ and also $(1+i)^8$ . So,as for the first one, I tried writing $2^{12}(\cos5 ...
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4answers
92 views

Math question please ? Complex numbers?

I have to solve this equation $5z^2+6z+2=0$ where $z$ is a complex number.. I tried writing $z=\alpha+\beta i$ but still nothing..I tried finding the roots but the discriminant is negative $= ...
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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.
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2answers
151 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 ...
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1answer
227 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 ...
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2answers
82 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
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2answers
177 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 ...
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1answer
99 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
4k 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 ...
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2answers
234 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 ...
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2answers
129 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 $\bar z$ ...
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1answer
79 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 ...
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2answers
576 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 ...
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2answers
80 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
140 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 ...
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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?
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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. $\operatorname{Im}((\sqrt2+3i)^2)$ $| ...
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1answer
77 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} ...
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2answers
112 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
60 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
217 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 ...
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1answer
120 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
90 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 ...
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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 ...
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1answer
275 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 ...
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1answer
87 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
3k 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
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2answers
299 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: ...
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4answers
826 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 ...
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1answer
94 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. ...
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1answer
177 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.
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2answers
1k 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.
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3answers
97 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 ...
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4answers
177 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 ...
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0answers
227 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 ...
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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 ...
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1answer
131 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 ...
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3answers
916 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
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2answers
61 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 ...
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2answers
429 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
192 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: $f\left(\displaystyle \frac{1}{n}\right) = 2^{-n}$ ...
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1answer
244 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
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3answers
146 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 ...
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2answers
779 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 ...