Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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2
votes
2answers
52 views

When $\cosh (z)=0$?

I'm studying complex analysis and I'm wondering about all complex values of $z$ that satisfy the equation: $$ \cosh(z)=0 \,\, . $$ Is there a smart way to show all values that vanish with the ...
0
votes
0answers
28 views

Derive the formula: $f(z)=2u(\frac{1}{2}z,\frac{1}{2i}z)-2u(0,0)$

Let $𝑢(𝑥, 𝑦)$ be a harmonic function which is the real part of a holomorphic function $𝑓 (𝑧)$, so that $$𝑢(𝑥, 𝑦)=\frac{1}{2}(f(z)+\overline {f(z)})$$ Argue that $\overline {𝑓(𝑧)} = ...
3
votes
1answer
63 views

Summation of the given series

Is there anyway to find the sum of: $\cos(A)+\cos^2(2A)+\cos^3(3A)+....$ upto 'n' terms. Actually original question was to find sum of : $\cos(A)+\cos(2A)+\cos(3A)+...$ upto 'n' terms and I found it ...
0
votes
1answer
22 views

Given complex $|z_1|=r_1,|z_2|=r_2,|z_3|=r_3$, when $|z_1-z_2|+|z_1-z_3|+|z_3-z_2|$has max and what is it?

$|z_1|=r_1,|z_2|=r_2,|z_3|=r_3$, when $|z_1-z_2|+|z_1-z_3|+|z_3-z_2|$has max and what is it? the question is from this question but no square. I had a geometric solution which is very long.(will ...
0
votes
2answers
27 views

Find the real and imaginary parts of the following.

$$\frac{z-a}{z+a}; a \in \Re$$ The part I'm confused about is the $a \in \Re$. I know that this means that $a$ is a real number (not imaginary), but then how do I interpret the addition/subtraction ...
1
vote
1answer
36 views

Write $e^{\ln(5)}i$ in polar and rectangular form

Is there something I'm missing? Below is my attempt, but I feel as though I might have missed something to learn about complex numbers when $r=e^n$. $$|ie^{\ln(5)}|=e^{\ln(5)}$$ ...
0
votes
1answer
50 views

$z^{19}=(-1+i)$ find the value of $z$. complex number

I have to find the value of $z$ which satifsfies this equation and which has the second smallest positive argument $\theta$, $0<\theta<2\pi$. I have to find $r$ and $\theta$. The answer I got ...
2
votes
0answers
30 views

Limit of $\exp(z^2)$ as $|z|$ tends to infinity

Let $g(z) = \exp(z^2)$ and $L$ a ray starting at the origin. Determine those $L$ along which $g$ has a limit (finite or infinite) as $|z|$ tends to infinity and $z ∈ L$. Find the value of the limit ...
0
votes
1answer
24 views

Complex number algebra, help

I have $ \lambda^{2} = +/- i \frac{\omega}{\nu} $ when I try and solve for lambda i dont get the same answer as my text book. the book $ \lambda =+/- \frac{1}{\sqrt{2}}(1 + ...
0
votes
2answers
41 views

Find all $z\in\mathbb{C}$ such that $e^z = 1$.

We write $z=a+ib$. Now, $$1 = e^z = e^{a+ib} = e^a e^{ib} = e^a(\cos b + i\sin b) = e^a\cos b + ie^a\sin b$$ We have $$1 = e^a \cos b \\ 0 = e^a\sin b$$ Now, I don't understand why it has to be ...
1
vote
1answer
18 views

Complex Numbers - Quartic

Find two distinct real roots of the equation $z^4-3z^3+5z^2-z-10$, and hence solve this equation completely. The problem is how do you find the two distinct real roots?
1
vote
1answer
25 views

Give the Taylor series for the following $f(z)$; also, find $f^{(100)}(0)$

$$e^{3z}$$ I'm not sure how to approach this complex number problem. I know that $$1+3 x+\frac{9 x^2}{2}+\frac{9 x^3}{2}+\frac{27 x^4}{8}+\cdots$$ is true for $e^{3x}$, but how does this apply to ...
0
votes
2answers
44 views

Complex Numbers: How do I know which plane to use?

I'm new to complex numbers and I want to know precisely when I need to use which plane for graphing and a general idea of what the plot would look like. $$z=a+bi$$ When it's in the form above, I ...
2
votes
0answers
45 views

What is the motivation behind the solution of this problem involving complex numbers?

The problem is Suppose for three distinct complex numbers $a, b, c$ such that $|a|=|b|=|c|>0$ all of the three numbers $a+bc, b+ac, c+ab$ are purely real. Prove that $abc=1$ By playing with ...
0
votes
0answers
12 views

Coordinates of symmetry of point with respect to line?

A, B and Z are three complex numbers, Pa, Pb and Pz their representation in the complex plane. what is the complex number that corresponts to the symmetry of Pz with respect to the line PaPb ? I ...
4
votes
3answers
45 views

Compute all roots of $(-8)^{\frac{1}{3}}$

$$(-8)^{\frac{1}{3}}$$ The problem states to compute all roots of the complex number above. Below is my attempt, but my inquiries are if I did it right and why it doesn't match Wolfram. Wolfram only ...
0
votes
1answer
23 views

Convert to Cartesian (rectangular) form

Convert the following to Cartesian (rectangular) form and provide a graph. $$e^{i7\pi /2}$$ The problem comes after a long series of similar problems. However, the noticeable difference with this ...
1
vote
0answers
24 views

Draw Regions On the Complex Plane that Satisfy this Relation

I'm looking to draw a region that satisfies the following: $$ Im\left(\frac{z-z_1}{z-z_2}\right)=0 $$ What I know so far is this: the expression as it's given is not of the form $ a + bi $, as ...
0
votes
2answers
70 views

Compute all the roots (complex number problem)

$$ (-1+i)^{\frac{1}{3}} $$ Below is what I've attempted, but I'm not 100% positive if it's right. Also, and more importantly, how do I know if I've computed ALL of the roots of a complex number? ...
2
votes
1answer
29 views

Plotting the graph of $\operatorname{Re}(z)<2$

$$\operatorname{Re}(z)<2$$ My idea was that, in the complex plane, the graph looks like a ray starting at the origin and extending up to $2$, but not including that point. Then there would have to ...
1
vote
2answers
37 views

Is this f(z) function analytic?

Is $f(z) = z^2\sin z$ an analytic function for $z \in \mathbb{C}$ ? $z = x + iy $ I really don't know how to split this at this format $f(x,y) = P(x,y) + iQ(x,y)$, so I can prove that if this ...
0
votes
0answers
18 views

Axis of a glide reflection

I am currently taking a gap year before starting university and am trying to get a head start by teaching myself some of the course content. As a result I have no one to ask and no solutions to check ...
2
votes
4answers
131 views

Solve for $z$ (complex numbers)

$$ z^3=i $$ The problem simply states to solve for $z$, but I know that there is some concept to be practiced here about the nth roots of unity. I'm just beginning to learn this concept so I didn't ...
2
votes
1answer
18 views

Polar form of a complex measurable function

Let $f:(X, \mathscr{A}) \to ( \mathbb{C}, \mathscr{B}(\mathbb{C}))$ be a measurable function. I need to show that there exists a measurable function $\theta: (X, \mathscr{A}) \to (\mathbb{R}, ...
0
votes
1answer
38 views

Trigonometric identity via complex exponential

Noting that $$\text{Re}[z_1z_2] = \text{Re}[z_1]\text{Re}[z_2]-\text{Im}[z_1]\text{Im}[z_2],$$ how can $$\cos(\alpha+\beta) = \text{Re}\left[e^{j(\alpha+\beta)}\right]$$ be expressed? Give ...
4
votes
1answer
61 views

Are there “3+ dimensional” complex numbers? [duplicate]

As an engineer, I learned a lot about how to use complex numbers. One way I have heard $i$, the unit complex number, defined is: It is orthogonal to the real number line. Because ...
0
votes
2answers
47 views

Calculate $\frac{1+i\tan \alpha}{1 - i \tan \alpha}$

I have been asked to calculate $\frac{1+i \tan \alpha}{1-i \tan \alpha}$, where $\alpha \in \mathbb{R}$. So, I multiplied top and bottom by the complex conjugate of the denominator: $\frac{(1+i \tan ...
1
vote
1answer
38 views

Eigenvectors of $\begin{pmatrix}6&2\\-10&-1\end{pmatrix}$ (linear equations with complex numbers)

I want to compute the eigenvectors of the matrix $$\begin{pmatrix}6&2\\-10&-1\end{pmatrix}$$ and thus far I got the eigenvalues $\lambda_{1,2}=\frac{1}{2}(5\pm\sqrt{31}i)$. However solving for ...
-1
votes
2answers
44 views

Complex equation $\sqrt{z^2}=z$ [closed]

I am trying to solve $\sqrt{z^2}=z$. It looks trivial, but I believe it is not all the complex number. For example $-i$ is not the solution to this equation. Can someone help me?
0
votes
1answer
20 views

Why is the sum of the first $k$ powers of a $k$-th primitive root $\varphi_k$ of $1$ always $0$?

Let $\varphi_k\in\mathbb{C}$ be a primitive root of $1$. It turns out, that $$\varphi_k^1+\ldots+\varphi_k^k=0\text{ .}$$ If I draw the roots for some fixed $k$, I can see that this seems evident. For ...
0
votes
1answer
13 views

Finding real part of an exponential with more than 1 term in the exponent.

z = $ \ sqrt2 \ + isqrt{2} \ w = \frac{pi}{4} + 2i $. Find Re{Ze^(iw)}. Now, i have simplified the exponential form to $\ 2 \ e^{( \frac{pi*i}{2} \ - 2)} $. But not quite sure how to derive the real ...
0
votes
2answers
35 views

Finding the angle of $-2i$.

Given $z = -2i$, I am to find the exponential form. Now, the radius $= 2$. The angle is derived as $\tan^{-1} \frac{y}{x}= \theta $ . $y$ and $x$follow the form $z = x + yi$. Now, given all this, ...
17
votes
8answers
531 views

Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?

I am an $8^{th}$ grader that is taking Algebra I. But nearly everyday I try to learn things outside of what I am learning in class. Quite a while ago I discovered that $e^{i\pi} = -1$. This ...
0
votes
2answers
44 views

De Moivres Theorem question and complex numbers

Question is: Find the cube root of $27 (\cos 30° + i \sin 30°)$ that, when represented graphically, lies in the second quadrant. I did this: ...
-1
votes
1answer
55 views

To Prove $\text{Arg}(\bar{z}) = -\text{Arg}(z)$ if $z$ is not real.

In complex analysis, To Prove $\text{Arg}(\bar{z}) = -\text{Arg}(z)$, is it enough to show that $\arg(\bar{z}) = -\arg(z)$? If not, what additional proof do I need?
0
votes
1answer
34 views

How do I begin to tackle this complex number problem?

Update: here is how the problem should have been read. Describe $z$ such that $$ |z+5|=3 $$ Thoughts? My attempt: Let $z=x+iy$ $$|(x+5)+iy|=3$$ $$\sqrt{(x+5)^2 +y^2}=3$$ Solving for $y$ yields ...
-1
votes
0answers
23 views

When does $z\bar{a} +a \bar{z} + b$ represent a line parallel to real axis given that $b \in \mathbb{R}$ and $a$ is a complex number?

When does $z\bar{a} +a \bar{z} + b$ represent a line parallel to real axis given that $b \in \mathbb{R}$ and $a$ is a complex number? $\operatorname{Re}(a)=0$ $\operatorname{Im}(a)=0$ ...
3
votes
0answers
29 views

Is it always possible to find the roots of $P(z)=az^4+bz^3+cz^2+bz+a$, where $a,b,c \in \mathbb{R}^*$, by first dividing both sides by $z^2$?

A classic way to solve quartics in the form $P(z)=az^4+bz^3+cz^2+bz+a$, if we know that the roots lie on the unit circle, is to divide both sides by $z^2$ and then use the fact that if $$z=\cos \theta ...
2
votes
1answer
452 views

Orthogonality in Complex Plane

This is probably trivial, but it's really bugging me. According to the inner product on $\mathbb{C}$ (i.e. $\langle z,w\rangle=z\overline{w}$), $\langle i, 1\rangle\neq 0$. I'd like to say that $i$ ...
2
votes
1answer
53 views

Real and imaginary part functions

How can real and imaginary part of $e^{ix}$ could be used in complex tasks. I am aware that sine is expressed as imaginary part and cosine as real part, but I am confused when there is some sort of ...
-1
votes
2answers
57 views

Solve for $x$ in $\frac{x}{\ln(x)}=a$. Why does Wolfram alpha use complex numbers here?

Is there any possible way of doing this without using complex numbers? And why are complex numbers used?
0
votes
1answer
26 views

Find a (finite) sum involving n-th root of unity

Evaluate $1+2\omega+3\omega^2+\ldots+n\omega^{n-1}$ where $\omega\neq 1$ is an n-th root of unity. My solution Let $S=1+2\omega+3\omega^2+\ldots+n\omega^{n-1}$ $\omega ...
3
votes
1answer
57 views

Why does rearranging Euler's identity in this manner result in a false statement?

I placed each of the following steps into Wolfram alpha after working it out in my head. All steps prior to the one marked with the (*) held true. $e^{i \pi} = -1$ $e^{2i \pi} = 1$ $\ln(e^{2i \pi}) ...
5
votes
2answers
75 views

If $\omega$ is an imaginary fifth root of unity, then $\log_2 \begin{vmatrix} 1+\omega +\omega^2+\omega^3 -\frac{1}{\omega} \\ \end{vmatrix}$ =

If $\omega$ is an imaginary fifth root of unity, then $$\log_2 \begin{vmatrix} 1+\omega +\omega^2+\omega^3 -\frac{1}{\omega} \\ \end{vmatrix} =$$ My approach : $$\omega^5 = 1 \\ \implies ...
2
votes
4answers
35 views

Complex number - wrong result at the end

I need to solve this: $$ \frac{i^4+3}{i-1}$$ On my book the result should be: $-2-2i$ but I get: $-1-2i$ and I do not understand where the error is. My steps: $$ \frac{i^4+3}{i-1} = ...
3
votes
3answers
2k views

Find a maximum of complex function

I am trying to find a simple method that does not use the tools of advanced differential calculus to find following maximum, whose existence is justified by the compactness of the close ball ...
2
votes
3answers
26 views

The point of intersection of the curves arg (Z-3i)= 3$\pi/4$ and arg (2z+1-2i) = $\pi/4$

Question The point of intersection of the curves arg (Z-3i)= 3$\pi/4$ and arg (2z+1-2i) = $\pi/4$ is :- a) 1/4(3+9i) b) 1/4 (3-9i) c) 1/2(3+2i) d) none
0
votes
2answers
55 views

Why is it ok to take square root of $|z|^2$, $z \in \mathbb{C}$? [closed]

Why is it ok to take square root of $|z|^2$, $z=a+ib \in \mathbb{C}$? Is it because $$|z|=\sqrt{a^2+b^2}$$ $$ \implies |z|^2=a^2+b^2 \in \mathbb{R}$$ and the square root of a positive real number ...
0
votes
2answers
44 views

Prove that all roots of $(z+1)^n = z^n$ lies on a straight line given that $n$ is a natural number for all $n \ge 2$

Question Prove that all roots of $(z+1)^n = z^n$ lies on a straight line given that $n$ is a natural numbers for all $n \ge 2$. Should i start the question assuming $z$ to be of the form $z= ...
2
votes
2answers
81 views

When does a cubic equation has two roots with same absolute values

Let $$z^3+bz^2+cz+d=0$$ be a cubic equation with complex coefficients. Suppose $z_1, z_2$ and $z_3$ are its roots. I need to find a condition on $b,c,d$ so that $$|z_1|=|z_2|.$$ How can I find such a ...