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

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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: ...
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1answer
25 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 ...
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1answer
25 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: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
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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 ...
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1answer
45 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 ...
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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 ...
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1answer
27 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.
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1answer
72 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
21 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 $z\in M_a$. ($\mathbb{C^*}$ ...
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28 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 ...
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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
52 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 ...
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1answer
31 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? [on hold]

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 ...
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3answers
65 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.
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1answer
27 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 ...
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0answers
27 views

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

$ 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 $x^n+y^n=i$ ...
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2answers
49 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 ...
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2answers
49 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.
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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 $$ ...
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2answers
55 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 ...
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1answer
27 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 ...
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0answers
16 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 ? ...
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
54 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 ...
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4answers
134 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.
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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 $f(z_1) = f(z_2)$ $\Rightarrow$ ${z_1}^2 +3z_1 +1= {z_2}^2 +3z_2 +1$ ...
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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 ...
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1answer
260 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: ...
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1answer
55 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
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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, $u$ ...
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1answer
19 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 $w+1=(1-b) + ai$ ...
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0answers
28 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 ...
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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 ...
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3answers
33 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, ...
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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?
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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[ ...
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1answer
49 views

$\cosh(iz) -\cosh(z)=0$

$\cosh(iz) -\cosh(z)=0$ Apparently $iz=z$ and $z=0$ is a solution. How do I proceed next? Do I need to convert $\cosh$ into $\exp$ form? I tried that I get $e$ to complex and real power and I don't ...
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2answers
63 views

How to solve inequalities with complex numbers such as $\operatorname{Im}(2/z)\geq13$? [closed]

How can you solve for $z$ (a complex number) an inequality such as $$\operatorname{Im}(2/z)\geq13 $$ and the second question is the inequality $$\frac{x^2-2}{2x+3}\geq x$$ I tried solving for both ...
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1answer
48 views

Prove the following identity in complex analysis (trigonometry) [closed]

We know $\cos(z)^2 + \sin(z)^2 = 1, \forall z \in \mathbb C$. Prove that, on the other hand, $|\cos^2(z)| + |\sin^2(z)| > 1, \forall z \in \mathbb C$ with $\text{Im}(z) \not = 0$.
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2answers
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arguments in complex numbers

I need to show that if $\arg\left(\frac{z_1+z_2}{z_1-z_2}\right)=\frac{\pi}{2}$ then $|z_1|=|z_2|$. How should I work this out? I know that $\arg\left(\frac{z_1+z_2}{z_1-z_2}\right) = ...
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1answer
29 views

Complex number forming equilateral triangle

Suppose we have 3 complex numbers , such that $$|z_1|=|z_2|=|z_3|=1$$ and they form equilateral triangle then will condition $$z_1.z_2.z_3=1$$ always be true? I know cube roots of unity , that is ...
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4answers
34 views

Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of y. Then indicate which one is the principle square root.

Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of $y$. Then indicate which one is the principle square root. Use the polar form of complex numbers to accomplish this task. I'm not ...
3
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0answers
28 views

Convergence of a complex function

I need to proof if the following function is bounded and convergent. $f(n)=\left(\frac{10+in}{n^{2}+2in}\right)^{n}$ Status: This should be correct. Can anybody confirm this? I tried it with ...
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2answers
34 views

Expressing complex numbers in standard form problem

Express the following complex number in the standard form $x + iy$ $ie^{\frac{i\pi}{2} +3} $ I have made an attempt and got the answer $\cos(\frac{\pi}{2} +3) +i\sin(\frac{\pi}{2} +3)$. Is this an ...
2
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1answer
14 views

Recursive sequence with Complex numbers, missing conclusion.

I am solving the following task: Let $a_1 = \sqrt{2}\sqrt{3}*i, a_{n+1} =\frac{i* a_n}{n+1}$ What can you say about the convergence of $a_n$? I already found out a lot. What i concluded so far, is: ...
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1answer
39 views

Values of the complex power $1^\sqrt{2}$

I have to show that the values of the complex power $1^\sqrt{2}$ all lie on the unit circle, i.e. that $|1^\sqrt{2}|=1$. $1^\sqrt{2} = e ^ {\sqrt{2} \ln{1}}$ by definition, and $\ln{1} = 2k \pi i$ ...