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

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3
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2answers
43 views

Using the polar form of $1 + i$ and $\sqrt3 + i$ to deduce $\cos (\frac{\pi}{12}), \sin(\frac{\pi}{12})$

I have been beating my head against the following problem and would like a gentle nudge in the right direction. The question states, by writing $1 + i$ and $\sqrt3 + i$ in polar form, deduce that ...
0
votes
2answers
42 views

Complex and Trigonometric Identities

How can I get this result: $$\frac{1+cis\theta}{1-cis\theta}=-\frac{1}{i\tan(\theta/2)}$$ I've tried to expand $1-cis\theta$ as $(1+cis(\theta/2))(1-cis(\theta/2))$, but it doesn't help.
0
votes
0answers
18 views

Complex number contour integral

Determine the contour integral ∫Ѱ 1/z dx, where Ѱ is the positively oriented unit circle with centre at -2 given that Ѱ(t) = -2 + e^(it), 0<=t<=2pii. I understand that Ѱ is the positively ...
1
vote
4answers
42 views

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$?

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$? Meaning, if I have a complex number and I multiply it by its complex conjugate, would that always return a number in $\mathbb{R}$? ...
2
votes
1answer
60 views

A problem on Möbius function

Here is an exercise from the text book "Combinatorics" (J.H.van Lint, R.M. Wilson) Let $f_n(z)$ be the function that has all its zeros as all the numbers $\alpha$ for which $\alpha^n=1$ but ...
2
votes
1answer
85 views

Does the partial order on $\mathbb{C}$ induced by $[0,\infty)$ have any non-trivial uses or application?

We can define a binary relation on the complex numbers as follows. $z \leq z'$ iff there exists $x \in [0,\infty)$ with $z+x=z'.$ Equivalently, $z \leq z'$ iff $\mathrm{Re}(z) \leq ...
1
vote
4answers
37 views

How do I reduce (2i+2)/(1-i) with step-by-step please?

I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the ...
0
votes
1answer
22 views

Behaviour of Hankel function $H_s^{(1)}(x)$ near $x=0$

I am looking for a reference to the fact $H_s^{(1)}(x) \approx i (\frac{2}{x})^s \frac{\Gamma(s)}{\pi}$ for small $x$,and $s\in \mathbb{C}$. I think it is obtained from some integral representation of ...
0
votes
0answers
45 views

Is derivation of Feigenbaum constant possible through Mandelbrot set?

this is Mandelbrot set: $z_{n+1}=z_n^2+C$ Is derivation of Feigenbaum constant possible through Mandelbrot set? $$\lim_{n\to\infty}\frac{z_{n+2}-z_{n+1}}{z_{n+1}-z_{n}}=\delta$$
0
votes
0answers
7 views

Calculating the DFT of a sequence following a mathematical expression

This is homework, so please don't give a full solution. Give a formula for $F_k$ for all $k$ where $f_n=4^n$ for all $n=0,\dots ,N-1$. I ended up abusing Wolfram Alpha and getting probably way ...
0
votes
0answers
32 views

Why are the real part and imaginary part of normal distribution function independent?

As I said in title, why are the real part and imaginary part of normal distribution function independent? I need a detail derivation to proof it. Thank you.
0
votes
0answers
5 views

How can we derive $\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)]$?

When I was reading digital communication theory, I couldn't derive following equation $$\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)] $$ ...
2
votes
3answers
170 views

solving complex numbers with powers algebraically

Find algebraically the value of :$\left(2^{0.5} + 6^{0.5} - \left( 2^{0.5} - 6^{0.5} \right)i \right)^4$ Below are my works I try to simplify inside. but i found that i can't add $2^{0.5}$ and ...
-1
votes
2answers
47 views

complex numbers algebraically

solve algebraically $x^2+2ix-5=3i$. My Solution: I tried using quadratic solution ( $a= 1 , b= 2i, c= -5-3i$ ) but it is wrong.
0
votes
1answer
56 views

If $|z|<1$ , show that $\left|\frac{1}{2}\arg (\frac{1+z}{1-z}) \right| < \frac{\pi}{2}$

If $|z|<1$ , show that $\left|\frac{1}{2}\arg (\frac{1+z}{1-z}) \right| < \frac{\pi}{2}$
1
vote
1answer
41 views

Proof complex series

I have to prove this: $\displaystyle\sum_{n=1}^\infty n\alpha^n = \displaystyle\frac{\alpha}{(1-\alpha)^2}$ if $|\alpha | < 1$ I think this is a geometric series, and i have to solve it with a ...
2
votes
2answers
90 views

Sum of $\sum\limits_{n=1}^\infty q^n \sin(nx)$

How to find $\sum\limits_{n=1}^\infty q^n \sin(nx)$, where $|q|<1$ and $x \in \mathbb{R}$? I was thinking about rewriting it as $\sum\limits_{n=1}^\infty (q(\Im(\cos x+i\sin x)))^n$. It is a ...
1
vote
1answer
53 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
1
vote
2answers
93 views

prove that there are no reals such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{x+y}$

I'm going through an introduction to complex analysis and there are two problems that I'm having problems with. A) Prove that there are no reals $x$ and $y$ such that $\displaystyle\frac{1}{x} + ...
2
votes
1answer
43 views

Neat way to prove $\sin(\alpha+\beta)$ using complex exponential

I am supposed to prove that $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ using complex exponentials: $$ \begin{align} \sin\theta&=-\frac{1}{2}i(e^{i\theta}-e^{-i\theta})\\ ...
1
vote
2answers
54 views

Need help solving this equation with complex numbers

$$(z^3 + 1)^3 = 1$$ where $z$ is an element of the complex number system. Can someone show me the most efficient way of finding all the solutions for $z$ here and also if possible please demonstrate ...
0
votes
4answers
40 views

Induction of logarithmic derivatives of complex functions?

I am trying to use induction to prove the logarithmic derivative of a complex function (called $P(Z)$ here). I define a function $P(z) = (z-z_1)(z-z_2)...(z-z_n)$ and then I want to use induction on ...
0
votes
2answers
61 views

How could I prove this trigonometric identity?

Show that: $$\left(\frac{1+\tan \theta}{1 - \tan \theta}\right)^n = \frac{1+i\tan n\theta}{1-i\tan n\theta}$$ Original image: http://i.stack.imgur.com/q8Yxj.jpg
0
votes
1answer
31 views

composition of complex functions

I'm given that $g(z)= \ln r+i\theta$ where $(r>0,0<\theta<2\pi)$ . I've already shown this function is analytic and that its derivative is $g'(z)=\frac{1}{z}$. Now it wants me to show ...
0
votes
2answers
38 views

Complex number multiplication

$z_1= \cos(4\pi/3) + i\sin(4\pi/3)$ $z_2= \cos(π/3) + i\sin (π/3)$ I want to find out $z_1z_2$. I know that $(x +iy)(u + iv$) = $(xu - yv) + i(xu + yv)$ So I want to simplify $\cos(4π/3)\cos(π/3) ...
0
votes
1answer
66 views

Is $\sqrt{-1}$ equal to $i$ or $\pm i$?

In complex numbers is $\sqrt{-1}$ equal to $i$ or $\pm i$ ? In both cases how do we explain it? The question arose when I saw it in Lathi's book (Linear Systems and Signals).
0
votes
2answers
47 views

Complex function mapping the unit circle onto an interval

Show that the function $f(z) = z^2 + z^{-2}$ maps the unit circle onto the interval $[-2, 2]$. Okay so far, doing previous questions I firstly try and find the inverse mapping. Here I considered the ...
1
vote
1answer
15 views

viewing complex numbers as a linear transformation

I am studying an intro to complex analysis and geometry book in order to become more adept with complex numbers and hopefully eventually the basics of complex analysis. I love the explanations but I ...
2
votes
2answers
55 views

Simplfy a complex matrix into a real one

I encounter systems of linear complex equations (At most 3 equations) in my circuit analysis course. The calculator I am using is Casio fx-991ES and it only accepts real elements when in matrix or ...
1
vote
1answer
27 views

Integral calculus use of Newton-Leibnitz rule

My friend asked me this question: If $y(x)= \int_{0}^{x}f(t)\sin{(px-pt)}dt$ then what is the value of $y''(x)-((p^2)*y(x))$. He gave me the hint to consider $\sin(px-pt)$ as the imaginary part of ...
0
votes
1answer
20 views

On the fundamental favor of usage of imaginary numbers over polar and spheric coordinates

So, I've been doing a little bit of research lately and stumbled upon two neat explanations, cases in favor maybe, of complex numbers. One said that complex numbers tried to explain rotation on a ...
0
votes
1answer
20 views

Prove that $2 \alpha/ n< \alpha^2 - t$

Hi : I am trying to learn about complex variables so I've been working my way through a book called An Introduction to Complex Analysis and Geometry by John D'angelo. It's a very nice basic book but ...
0
votes
2answers
28 views

How to plot a curve in complex plane that include e constant

I can understand how a complex numbers such as $10 + 7i$ can be plotted in complex plane with Imaginary and Real axis. But I have no idea how to approach this problem. I have been asked to plot ...
0
votes
0answers
10 views

Some kind of regression on complex numbers

I need to compute (or at least tell something about) an expression like this: $$\sum_{k=1}^n ||\rho e^{i(t_k\omega-\theta)}-\sum_{j=1}^p \rho_j e^{i(t_k\omega_j-\theta_j)}||^2$$ The idea is to find ...
0
votes
1answer
34 views

Find the radius of convergence of the power series

$\displaystyle\sum_{n=0}^{\infty}a_nz^n$, where $a_{2k+1} = 2^k$ and $a_{2k} = (1 + (1/k))^2$ for $k = 0, 1, 2, \dotsc$ I started off by doing the ratio test, but I know that the ration test is for ...
0
votes
0answers
23 views

Evaluate $\int f(z)\,dz$ with $f(z) = x^2 - iy^2$, where $z = x + iy$, and the curve $C$ is given by $C(t) = t - it^2, 0<t<1$

I began by assigning the $\Re z$ to be $x = t$ and the $\Im z = y = -it^2$. Then I computed $z'(t) = 1 - 2ti$. Then $f(z(t)) = t^2 - it^4$. Then I took $$ f(z(t)) z'(t) = (t^2 - it^4)(1-2ti) = (t^2 ...
1
vote
2answers
34 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
0
votes
1answer
33 views

how to solve this problem on complex analysis

in a probelm set I found $(x + \sqrt2j)(x − \sqrt2ij) =(x^2 + 2x + 2)$ where $j=√i$ but i can't understand how this happened. I have done this $(x + \sqrt2j)(x − ...
12
votes
2answers
628 views

Prove that a polynomial has at least one nonreal complex root

Prove that the polynomial below has at least one nonreal complex root $$x^5+\frac{x^4}2+ \frac{x^3}3+\frac{x^2}4+\frac x{24}+\frac 1{120}$$ I have tried to prove that there exist $k\in \Bbb R$, such ...
1
vote
6answers
129 views

Why would you define $i$ as $i^2=-1$, and why not as $i=\sqrt{-1}$?

I'm not sure if I recall this correctly, but I thought there was a reason why you shouldn't write $i=\sqrt{-1}$. And if this is not true, then I wonder: Why would you define $i$ as $i^2=-1$, why ...
1
vote
3answers
31 views

solving complicated complex numbers

Find values for $a$ and $b$ so that $z=a+bi$ satisfies $\displaystyle \frac{z+i}{z+2}=i$. Below are my workings: so far i simplify $\displaystyle \frac{z+i}{z+2}=i$ to $z=zi+i$ which $a=i$, $b=z$
1
vote
1answer
41 views

Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
-1
votes
1answer
34 views

Calculate the following expression value (Complex Numbers)

Hi the following is the equation to solve which I am lost as I do not quite understand the method of using complex numbers to solve. Please help! $ \sqrt[4]{\frac{-18} {1+i\sqrt{3}}} = \frac ...
0
votes
2answers
33 views

Imaginary Number Adding One Explanation

On Khan Academy there is this question. $$0.33i−i^3$$ I don't understand how the answer is: $$0+1.33i$$ Could someone please explain why it's adding $1$ to the $0.33$?
2
votes
1answer
34 views

How to find out the real part of this expression in term of k.

$$A=\frac{e^{i \pi k}-1}{e^{i \pi k h}-1}$$ Where $k=p+q$, $p$ and $q$ natural numbers $h\in]0,1[$ real number We can consider that the denominator is never $0$. The result may be someway like ...
1
vote
3answers
37 views

Real part of a quotient

Is there some fast way to know the real part of a quotient? $$\Re\left(\frac{z_1}{z_2}\right)$$ $z_i\in \mathbb{C}$
2
votes
2answers
32 views

complex numbers for quadratic equation

Find the quadratic equation whose roots are $2+i$ and $3-i$. $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
4
votes
2answers
181 views

Finding the roots of the sum of complex numbers

Find the roots of $(z-1)^6 +(z+1)^6$. So far we've tried binomial expansion, but where to go now, as it is a non-calculator question?
1
vote
1answer
44 views

Finding all roots in a complex system/equation

I dont understand where to begin, or how to approach this question. it asks: find all the roots of: $$(1 + \sqrt{3}i) ^{1/2}$$ should I put it into polar form first? $$z = re^{ix}$$ what throws ...
1
vote
1answer
55 views

Unit circle traversed once in the punctured plane not homotopic to the same circle traversed twice.

On page 185 of Saff, Fundamentals of Complex Analysis, the author states that in the punctured plane ($\mathbb{C}- \{0\}$), the unit circle traversed once in the positive direction is NOT continuously ...