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

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4
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1answer
138 views

When $|f|$ and $\arg(f)$ are analytic?

Let $f:ℂ→ℂ$ be an analytic function. Define $|f|$ and $\arg(f)$ be the modulus and the argument of $f$. Generally, $|f|$ and $\arg(f)$ are not analytic. My question is about the cases where this ...
-2
votes
4answers
102 views

Complete instead of Complex, Irregular instead of Imaginary

Will the terms complex and imaginary ever be replaced? At least within beginning classes? I imagine it is more of a kind of hazing into the "mathemitician's club" to allow the terms to confuse ...
1
vote
0answers
49 views

show this summation hold in term of integral

i can show $\sum |x|^2=\int_a^b|f(x)|^2dx$ in term of integral, or this one $|\sum x\overline y|^2=|\int_a^bf(x)\overline {g(x)}dx|^2$ but i don't know how to show this one ...
1
vote
1answer
69 views

What is more compact equation of this relationship?

What is more compact equation of this relationship? $\sum |x_i|^2\sum |y_j|^2+\sum |x_j|^2\sum |y_i|^2-2|\sum x_i \overline y_i||\sum x_j \overline y_j|$ Remark: Euclidean space $\sum x_i^2\sum ...
2
votes
4answers
89 views

Motivating complex structure on $\mathbb{R}^2$

I'm giving a talk to a group of bright but not all that mathematically sophisticated students on the subject of complex numbers. I'd like to introduce complex numbers via geometric considerations ...
18
votes
4answers
772 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
0
votes
1answer
236 views

Complex form of gauss divergence theorem

Just as complex form of green's theorem $\int {f(z)}dz=i\int\int \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}dxdy$ where $z=x+iy$ , do we have complex form of gauss divergence ...
2
votes
2answers
145 views

upper and lower bounds of a complex expression

How do I prove that $$\sqrt{\frac{7}{2}}\leq |1+z|+|1-z+z^2|\leq 3\sqrt{\frac{7}{6}}$$ for all complex numbers $|z|=1$? I don't really know how to grapple with it. P.. I am extremely sorry, the ...
3
votes
1answer
2k views

Relationship between complex number and vectors

What is the relation between complex numbers and vectors? I have read in some places "a complex number a 2-dimensional vector". One can easily observe that $i\cdot i=-1$ in complex multiplication ...
3
votes
1answer
251 views

calculation proof of complex form of green's theorem

Complex form of Green's theorem is $\int _{\partial S}{f(z)\,dz}=i\int \int_S{\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\,dx\,dy}$. The following is just my calculation to show both ...
3
votes
4answers
108 views

condition for two complex number $a$ and $b$ to be $|a+b| = |a| + |b|$

If $a$ and $b$ are two complex numbers, and $a \neq 0$, then how to show that the condition required for $|a+b| = |a| + |b|$ is $b/a$ is real and non-negative. I did the following and I got stuck ...
1
vote
1answer
98 views

Show That This Complex Sum Converges

For complex $z$, show that the sum $$\sum_{n = 1}^{\infty} \frac{z^{n - 1}}{(1 - z^n)(1 - z^{n + 1})}$$ converges to $\frac{1}{(1 - z)^2}$ for $|z| < 1$ and $\frac{1}{z(1 - z)^2}$ for $|z| > ...
2
votes
2answers
148 views

Complex Eigenvectors

This is a relatively simple question and I've Google'd this topic but I can't seem to grasp the method. One site I visited helped be a bit, but it simply used substitution to solve the problem rather ...
2
votes
3answers
239 views

How to calculate $\overline{\cos \phi}$

How do you calculate $\overline{\cos \phi}$? Where $\phi\in\mathbb{C}$. I try to proof that $\cos \phi \cdot \overline{\cos \phi} +\sin \phi \cdot \overline{\sin \phi}=1$?
2
votes
2answers
81 views

Eigenvalues of a certain bordered identity matrix

Consider a complex $N-1 \times 1$ vector $b$ and a complex constant c. Let $I$ denote the $N-1 \times N-1$ identity matrix. Then what can we say about the eigenvalues of the matrix \begin{align} ...
0
votes
3answers
367 views

Real and Imaginary Part of Large Power

I came across this problem in a complex analysis book: Find the real and imaginary parts of $(1 + i)^{100}$. Now, this question is asked before polar form is introduced, so I am curious about ...
2
votes
1answer
134 views

complex numbers

I know that $f$ is continuous on $[a,b]$ with $ab\neq0$, $f(a)f(b)\neq0$ and the complex numbers: $$ z = a^2 + f(a)i $$ $$ w = b^2 - f(b)i $$ $$|\bar w + z| = |w - \bar z|$$ 1)Prove that $w\cdot z$ ...
1
vote
1answer
137 views

Value of $x$ & $y$ in computing gabor filter function?

I have trouble understanding in an intuitive way (not by writing complicated math formulas) what is value of $x$ & $y$ in the Gabor functions. Here is the formula, $$g(x,y) = ...
3
votes
3answers
138 views

How to find $(-64\mathrm{i}) ^{1/3}$?

How to find $$(-64\mathrm{i})^{\frac{1}{3}}$$ This is a complex variables question. I need help by show step by step. Thanks a lot.
0
votes
3answers
227 views

How to figure out the principal argument for z?

If $z = (4\sqrt{3} - 4 i)^3$, determine $\arg z$. How to find out this $\arg z$? i need help. thanks a lot...
1
vote
1answer
65 views

Find the Möbius transformation mapping $(i, 0, \infty)$ to $(0, \infty, -i)$, in precisely that order

I did this: Assuming my Möbius transformation is some $\omega$ in terms of $z$, I want to work out a formula that gives me: 1) $\omega = 0$ when $z = i$ 2) $\omega = \infty$ when $z = 0$ 3) $\omega ...
1
vote
1answer
92 views

Representing roots of unity

Is there some notation in terms of $n,k$, I can use to represent the complex exponential $e^{2\pi i\frac{k}{n}}$, I find by writing the exponential out, I often make mistakes and it is timely to write ...
1
vote
3answers
512 views

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 ...
1
vote
2answers
114 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 ...
7
votes
2answers
618 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 ...
3
votes
3answers
3k 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.
2
votes
1answer
269 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 ...
0
votes
1answer
96 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 ...
-2
votes
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 ...
2
votes
4answers
91 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 $= ...
0
votes
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
votes
2answers
142 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
votes
1answer
218 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
votes
2answers
79 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
votes
2answers
165 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
vote
1answer
93 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
45
votes
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 ...
1
vote
2answers
215 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
votes
2answers
122 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
vote
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 ...
1
vote
2answers
540 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
vote
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 ...
-2
votes
1answer
136 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
votes
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
vote
2answers
108 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}) + ...
1
vote
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!
0
votes
2answers
192 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
119 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 $r$?