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

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4
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0answers
81 views

How can $ i $ be distinguished from $ - i $? [duplicate]

Mathematicians designate one solution to $x^2 = -1$ as $i$ and the other as $-i$. Would anybody notice if we switched their identities? Any polynomial $p(x)$ with a complex root will also have its ...
1
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1answer
27 views

Images of lines $y = k = \mbox{constant}$ under the mapping $w = \cos (z)$

I want to solve this question: find the images of lines $y = k = \mbox{constant}$ under the mapping $w =\cos(z).$ I know that $w=\cos(z)=\cos(x)\cosh(y)-i\sin(x)\sinh(y)$ so $u=\cos(x)\cosh(y)$ and ...
0
votes
3answers
44 views

Derivative of a quadratic form

There is a Hermitian matrix $X$ and a complex vector $a$. I know that $a^HXa$ is a real scalar but derivative of $a^HXa$ with respect to $a$ is complex, $$\frac{\partial a^HXa}{\partial a}=Xa^*$$ Why ...
0
votes
1answer
27 views

Checking where the complex derivative of a function exists

I have the following function: $$f(x+iy) = x^2+iy^2$$ My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we ...
0
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1answer
41 views

How to build $\mathbb{C}$

I've defined $\mathbb{C}$ as $\mathbb{R} [X]/ (X^2+1)$, how do I show that $\mathbb{Q} [X]/ (X^2+1)$ is a subset of $\mathbb{C}$? And is $i \in \mathbb{Q} [X]/ (X^2+1)$? And can we see $\mathbb{Q} ...
3
votes
1answer
47 views

Drawing complex numbers on an argand diagram

I need some help drawing the following loci (which are rather hard to comprehend for me how will they look like) on an argand diagram: $$\arg \frac{i-z}{z+i}=\frac{\pi}{2} $$ (this one I suppose is ...
0
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1answer
37 views

Quadratic formula for $z^2 + (\alpha + \beta i)z + \gamma + i\delta = 0$ where $z\in\mathbb{C}$

The problem statement is to solve the quadratic equation $$ z^2 + (\alpha + i\beta)z + \gamma + i\delta = 0. ...
0
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2answers
24 views

Looking for proof of theorem on complex measurable functions

In University I have been given the following result: If $f:X\to\mathbb{C}$ is a measurable function in $L^1(X,\mathcal{E},\mu)$ with $\mu$ being finite, and there exists a closed set ...
0
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0answers
25 views

Is split-complex $j=i+2\epsilon$?

In matrix representation imaginary unit $$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$ dual numbers unit $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ ...
6
votes
3answers
128 views

How to visualize $f(x) = (-2)^x$

Background I teach Algebra and second year Algebra to middle school students. We are currently studying Exponential, Power, and Logarithmic functions. We study exponential functions (of the form ...
0
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0answers
28 views

Is restriction of $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

I have this question: Is the restriction of exp function $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection? Here's what I tried: ...
1
vote
1answer
28 views

Solution for a complexed equation

Find $z$ for the equation $e^z + e^{-z} = 0$. So $$e^z + e^{-z} = 0 \iff e^z = -e^{-z} \iff e^z = e^{\pi i - z} \iff z = \pi i -z + 2\pi ik$$ I understand all expect the $2\pi ik$. Can you ...
1
vote
1answer
26 views

find roots in the complexes

Find the roots of: $$ z^2 -3z +4iz = 1-5i $$ Rearranging the terms: $z^2 + z(4i-3) + 5i - 1 $ Solving by using the quadratic formula: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
0
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0answers
19 views

Does $t^2y''-ty'+y=2t$ with $y(t)=tz(t)$ only if $z'$ as $t^2u'+tu=2$?

I have this question: We have $y$ and $z$ functions with reals as $y(t) = t\times{z}(t)$ for $t \in I = ]0,+\infty[$. Then $y$ satisfies $t^2\times{y}''-t\times{y}'+y=2t$ on $I$ if and only ...
0
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1answer
48 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
0
votes
1answer
30 views

Are $\Re(z)$ and $\Im(z)$ solutions of $z' = az$?

I'm having trouble with a question (I have to answer "true" or "false" and explain it): We have $a:I \to C,$ a continuous function on $I$, interval of $R$. If the function $z:I \to C$ is a ...
3
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1answer
28 views

Is i holomorphic over the whole complex plane?

That is, is i entire? I know that it's derivative with respect to z bar is 0, so I would think that the answer is yes, although I'm not sure.
7
votes
1answer
77 views

Does $\exp(2ir\pi)$ equal $1$? What's wrong?

Since $e^{ix}=\cos x+i\sin x$, thus $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$ Now I take arbitrary real number $r$ then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$ But this cannot be true since $\cos2\pi ...
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0answers
22 views

Finding extreme complex numbers satisfying a condition

Let $a$ be a positive real number and let $$M_a = \left\{z \in \mathbb{C^*}: \left|z + \frac{1}{z}\right| = a\right\}$$ Find the minimum and maximum value of $|z|$ when $z\in M_a$. ($\mathbb{C^*}$ ...
0
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0answers
32 views

Chain rule (derivative) for for complex data

I found some difficulties in extending the chain rule for complex data. Any suggestion will be appreciated, thanks. In the complex domain, for example, we have a function ...
0
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2answers
35 views

to find radius of convergence of power series.

I have a power series given as: $f(z) =1 + z+ \frac{z^2}{2^2} +\frac{z^3}{3!} + \frac{z^4}{2^4} \frac{z^2}{2^2}+ \frac{z^5}{5!}+ \ldots$ I have to find radius of convergence of above series. My ...
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1answer
53 views

how to solve complex integration problem

While working on complex integration problem I got stuck at the following problem: $\int \frac{|dz|}{|z-2|^2}$ where $|z| = 1$ is the domain. The only idea that I am getting is that we can use the ...
0
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1answer
32 views

Complex roots of Complex polynomal

Apologies if this is a repeated thread I just couldn't quite find anything that helped. how do I go about finding the complex roots of a complex polynomial? such as $$x^3 + (1-i)x^2 + (1-i)x - i$$ ...
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3answers
43 views

Describe the solutions of the equation in terms of roots of unity? [closed]

I want to find the solutions of the equation $$\left[z- \left( 4+\frac{1}{2}i\right)\right]^k = 1 $$ in terms of roots of unity. When I try to solve this, I get \begin{align*}z - 4 - \dfrac i2 ...
2
votes
3answers
68 views

Find the set of complex numbers $z$ which satisfy: $\left\lvert\frac{z-3}{z+3}\right\rvert=2$

Find the set of complex numbers $z$ which satisfy $$\left\lvert\frac{z-3}{z+3}\right\rvert=2\text.$$ I need help on that one. Thank you.
0
votes
1answer
29 views

Magnitude of a complex expression

Is there a way to derive an expression for the magnitude of $$ \frac{2 + (1-2ia\lambda \sin \theta)^{1/2}}{3 + 2ia\lambda\sin\theta} $$ I know how to do this if the square root weren't there. Any ...
0
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0answers
28 views

For which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$? [closed]

$ x $ and $y$ are real numbers and $i$ : is unit imaginary part . 1-for which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$ ? 2-what are the possible geometrics forms of $x^n+y^n=i$ ...
3
votes
2answers
50 views

Proving that for all complex $z$, $\lim_{x\to0}\frac{1-\cos^{z}x}{x^2}=\frac{z}{2}.$

What do I need to study beforehand in order to prove it (not necessarily in only one way)? I found this sperimentally, at the moment we're beginning derivatives at school. By induction, I succeeded in ...
1
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2answers
50 views

Find all solutions for $z^3 = \overline{z}$

I know that $z = a + ib$ and that $\overline{z} = a - ib$, but when I try and calculate the solutions I get an unsolvable equation. Would appreciate any help.
2
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3answers
65 views

If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$

Question : If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$ What I did : $$|z-3i| \leq |z|+3 \tag i$$ Also $$|z-4| \leq |z| +4 \tag{ii}$$ Now adding (i) and (ii) we get $$ ...
0
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2answers
56 views

How to change the limits of a summation when the index $k$ is replaced by $-k$?

Is what I am doing below correct, please assist. $$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ Is this the rule on how to "invert" the limits, and does ...
1
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1answer
29 views

Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ …

Problem : Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ My approach : Let $z = x +iy$ $ \Rightarrow (x^2+y^2)=1$ $\Rightarrow |z| =1 $ represent a circle with centre at (0,0) and radius 1 ...
0
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0answers
17 views

Diffferentiability of complex functions

I need you help me please. I don't know how solve this Find $f_{z}$ y $f_{\bar{z}}$ where $f(z)=\left |{z}\right |^{2} +\displaystyle\frac{z}{\bar{z}}$ moreover what points is differentiable f ? ...
0
votes
1answer
31 views

inequality in absolute value with exponential

Can you help me with this problem please ?? If $r>0$ . Show that $\left |{\displaystyle\int_{\gamma}e^{iz^{2}}dz}\right |\leq{\displaystyle\frac{\pi(1-e^{-r^{2}})}{4r}}$ where ...
0
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1answer
41 views

how to prove that a function is not complex differentiable

I was working on a problem on the complex differentiability of the following function: $f(z)= z \operatorname{Re}(z)$. How to find the points where the given function is not differentiable. My ...
1
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1answer
25 views

How to find $\lim\limits_{z \to z_0} \frac{{\overline z}^2-{\overline z_0}^2}{z-z_0}$

Fairly simple question, I want to find this limit $$\lim_{z \to z_0} \frac{{\overline z}^2-{\overline {z_0}}^2}{z-z_0}$$ The original question was to find the region at which the function ...
1
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4answers
37 views

Can we add fractional powers of negative numbers?

This question might be silly and very basic. But my friend and me happened to argue on this for long. My argument was, if $-2 \sqrt3=\sqrt{12}$ which came from $\sqrt{(-2)(-2)} \sqrt{3} $ . If this is ...
0
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0answers
55 views

Finding complex eigenvectors of $n \times n$ matrix, $n\geq 3$

An example: $$ \begin{pmatrix} 1 & 2 & 0 \\ 2 & -3 & 4 \\ 4 & -8 & 7 \\ \end{pmatrix} $$ Has eigenvalues $3$, $1+2i$, $1-2i$. How ...
3
votes
4answers
135 views

Minimum value of $|z+1|+|z-1|+|z-i|$

How to find the minimum value of $|z+1|+|z-1|+|z-i|$. I have tried geometrically etc but failed.
0
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1answer
10 views

how to prove that a given function is univalent

I have to prove that following function is univalent $f(z) = z^2 +3z +1, ~|z|<1$ in complex plane. What I tried is: Let $f(z_1) = f(z_2)$ $\Rightarrow$ ${z_1}^2 +3z_1 +1= {z_2}^2 +3z_2 +1$ ...
1
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2answers
49 views

Solving $e^z=\pm1$ in $\bf C$

given $e^z = -1$ we have: $e^z = 1*e^{i\pi}*e^{2ki\pi}$ and taking $ln$ both sides yields: $z = ln(1)+i\pi+2ki\pi$ and if given $e^z = 1$ we have : $e^z = 1*e^{2ki\pi}$ and taking $ln$ both sides ...
4
votes
1answer
262 views

When was it realized that complex numbers can't lie on a number line?

When I first learned about representation of a complex number by a point in a $2D$ plane, I wondered: what if it's redundant? What if a line is sufficient? Apparently, it's not, but I still wonder: ...
2
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1answer
56 views

Can someone please explain the following definion of $\ln(e^z)$

I noticed someone do this from one of the questions is asked on here i had: $$e^z = -0.5$$ $$e^z = 0.5e^{i\pi}$$ which magically became: $$z = \ln\left(\frac12\right) + iπ + 2ikπ$$ does this mean ...
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0answers
26 views

Urgent quick harmonic function questions [closed]

Straight to the point; A function can only have one harmonic conjugate - am I right in saying that? How do you test a function is holomorphic/analytic? can I prove if, for a particular case, $u$ ...
1
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1answer
20 views

Sketching complex numbers on an Argand diagram help

for {${z\in \mathbb C : Im(z)>0}$}, we simply sketch the upper half of the Real axis, right? Then, if we have $z=a+ib$, and we sketch that, and we have $w=iz=-b+ai$ which means $w+1=(1-b) + ai$ ...
1
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0answers
29 views

Notion of complex optima

Consider the function: $$y = \frac{1}{3}x^3 + x$$ Suppose we wanted to determine its local optima, but instead of looking at local optima with domain $R$ we instead consider domain $C$ and range ...
0
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1answer
35 views

Prove that if a complex number raised to nth power equals one, then product of the solutions to this euqations is one

I've thought of this "proof", but it seems lame to me (I'm not sure it is even a proof), I probably should have used other properties of complex numbers to write one. So, $z^n=1 \implies ...
2
votes
3answers
34 views

Solving $z^3=-1+i$

First, is there a better way than using x+iy and solving the system? I tried letting $z=e^{i3\theta}$ and using the cosine and i*sine way but I don't see how that can equal -1 and i at the same time, ...
0
votes
4answers
48 views

argument of the complex number $1-\cos x-i\sin x$

How can I find the argument of $1-\cos(x)-i\sin(x)$? Can I use the exponential form of a complex number? If so, how do I continue to find the the answer?
1
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1answer
27 views

Complex matrix calculations

Sorry about the vague subject but I really found some difficulties in calculating complex matrices. Assume $Z$ is a square Hermitian non-singular complex matrix, then we denote $$F= \left[ ...