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

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0
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1answer
17 views

Orthogonal set of complex functions

Show that the functions $e^{in\pi x/l}$, n = 0, ±1, ±2, ..., are a set of orthogonal functions on $(-l, l)$ using: $A(x)$ and $B(x)$ are orthogonal on $(a,b)$ if $\int^b_a A^*(x)B(x)dx = 0$ ...
0
votes
2answers
28 views

Solution to some confusing complex equation

I am asked to solve the equation: $(4-3i)z^2-25z+31-17i = 0$ I approached it this way: $(4-3i)$[$z^2$-$(25z/((4-3i))$+$(31/(4-3i))$-$(17i/(4-3i))$]=0 which led to ...
0
votes
1answer
20 views

Proving the range of $Log(z+i)$

I have the function $Log(z+i)$ and I am told the range is $$ R=\{x+iy : 0<y<\pi,\;\; x<\log(2\sin y)\} $$ How do I go about showing that this is true?
0
votes
0answers
15 views

Why are these triangles formed by the product of two complex numbers similar?

I was trying to understand Eulers formula from this link and I came across this image on the second slide: I'm trying to understand why the specified triangles are similar. One intutive ...
0
votes
2answers
36 views

Prove a complex inequality

I need to prove that \begin{gather*} | z^4 + z + 1 | \geq |z^4| - |z| -1 \end{gather*} with $z = (r\exp (it))$ and $r \geq 2^\frac{1}{3}$ It seems like I only have to use the reversed triangle ...
2
votes
3answers
244 views

What is the geometric interpretation of the following equation?

What is the geometric interpretation of the following equation? $\displaystyle\left|\frac{1+z}{1-i\bar{z}}\right|=1$
1
vote
1answer
46 views

Find square roots of $8 - 15i$ [duplicate]

Find the square roots of: $8-15i.$ Could I get some working out to solve it? Also what are different methods of doing it?
1
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4answers
63 views

Find $a$ and $b$ when $(a + ib)^2 = 2i$ [closed]

Find the values of $a$ and $b$ in the following: $$(a + ib)^2 = 2i$$ Could I possibly get the working for this? Thanks guys!:)
0
votes
2answers
24 views

How to write $w^3$ as $a+bi$? [closed]

If $w = 3 + i$, write $w^3$ in the form of $a+bi$. Please help, thanks!(:
0
votes
2answers
35 views

To prove $\sum\limits_{k=1}^{n}k \cos{\frac{2k\pi}{n}} =-\frac {n}2$

Prove that $\sum\limits_{k=1}^{n}k \cos{\frac{2k\pi}{n}} =-\frac {n}2$, for $n\in\mathbb{Z}, n\ge3$
0
votes
1answer
18 views

Integral $\cos ^+ (\alpha - \phi)$

In Rudin's Real and Complex Analysis there's a proof of the statement that If $z_1 ..., z_N$ are complex numbers then there is a subset $S$ of $ \{1,..., N \}$ for which $|\sum_{k \in S} z_k| \ge ...
-1
votes
2answers
47 views

least value of a complex number

If $z_1,z_2,z_3,z_4\in C $ satisfy $z_1+z_2+z_3+z_4=0$ and $|z_1|^2+|z_2|^2+|z_3|^2+|z_4|^2=1$ then what will be the least value of $|z_1-z_2|^2+|z_1-z_4|^2+|z_2-z_3|^2+|z_3-z_4|^2$? What approach ...
-1
votes
2answers
54 views

Root of a complex number [closed]

Please help me Evaluate the following and display on an Argand Diagram. $z^{4/5}$ given $z=-1-j$ Thank you.
0
votes
2answers
42 views

Parameterize $ \left| z - 2 \right| = 1 $.

Parameterize $$ \left| z - 2 \right| = 1. $$ I know that the answer should be $$ z(t) = 2 + e^{it}. $$ But I don't know how to do this. And no, I haven't triend anything since I don't know where to ...
1
vote
1answer
37 views

Sums of complex numbers - proof in Rudin's book

I have one question about a proof in Rudin's book: If $z_1 ..., z_N$ are complex numbers then there is a subset $S$ of $ \{1,..., N \}$ for which $|\sum_{k \in S} z_k| \ge \frac{1}{\pi} \sum_{1}^N ...
0
votes
3answers
27 views

Problems while solving equation with $e^z$

I'm trying to solve this equation: $e^z-2ie^{-z}=i-2$ Here's what I've done: $z=x+iy$ $e^x(\cos y + i \sin y) - 2ie^{-x} (\cos y - i \sin y) = i-2$ (1) $ \ \ e^x \cos y - 2e^{-x} \sin y = -2 \ $ ...
0
votes
1answer
26 views

complex numbers find greatest value of z

I've to sketch the complex number $z$ such that it satisfy both the inequality $|(z-2i)|\le2$ and $ 0\le \arg(z+2)\le 45\deg $ I was able to sketch and shade the region that satisfies both ...
-1
votes
1answer
29 views

What is the argument of the root with the greatest argument? [closed]

Can you please help me solve the equation $$z^{6} = 5-4i$$ for $z$? I will be really thankful if you can, thanks.
0
votes
1answer
14 views

Linear independence of a complex function

I need some help to show that $x_1(t)=(e^{\lambda t})u$ and $x_2(t)=(e^\bar {\lambda t})\bar u$ are linearly independent complex solutions of the homogeneous equation $x'(t)=Ax(t)$ where $A$ is a ...
2
votes
3answers
41 views

Linear Algebra - Complex equation

I have this problem : $$x^2=i$$ The args = $\pi/2$. $r = |z| = \sqrt{0^2+i^2}=\sqrt{i^2}=i$ for $$z_0=i((\cos (\pi/2)/2)+isin(\pi/2)/2)) = ...
-5
votes
3answers
39 views

Loucs in complex plane [closed]

Describe the set of points in complex plane that satisfy the given equation $$Re\ \bigg(\frac {z+j}{z-j}\bigg)=1$$
0
votes
2answers
36 views

Finding maximum value of absolute value of a complex number given a condition.

On a recent test, I could not solve the following problem: If $$\left | z^2 + 2zcos\alpha \right | \leq 1 $$ then find the maximum value of absolute value of z. Alpha is not a fixed parameter. Alpha ...
0
votes
0answers
30 views

How to solve radical equations without creating any extraneous solutions while solving.

How does one solve a radical equation, such as $\sqrt{z+c_1}=z+c_2$, where $z \in \mathbb{C}, $ without creating any extraneous solutions at all while solving the equation. I know that by squaring an ...
1
vote
1answer
38 views

$\sum _1 ^n |z_j| \ge 1 \Rightarrow | \sum _1 ^k z_{j_m}| \ge C$

Prove that there exists $C > 0$ such that the following implication holds: If $\{z_1, ..., z_n \} \subset \mathbb{C}$ are such that $\sum _{j=1} ^n |z_j| \ge 1$, then there exists $ \{z_{j_1}, ...
1
vote
2answers
33 views

solve equation $z^4=7/4+6i$ complex number equation

Please help me to solve the equation $z^4=(7/4)+6i$ numbers are pretty complicated so I dont know how to solve it without using calculator and with exact number.
2
votes
2answers
75 views

How to solve the equation $\exp(iz)=-e$

How do I solve the equation $$\exp(iz)=-e$$ Can anyone please explain the procedure to solve this kind of question to me please? Much appreatiate
0
votes
1answer
28 views

A simple geometric method for finding the square roots of a complex number

Consider the following method for finding the $2$ square roots of a complex number: Draw the number on an $XY$ plane, as a vector starting from $(0,0)$ Let $L$ denote the length of that vector Let ...
1
vote
4answers
114 views

What is the square root of “i”?

Where i is the square root of negative one. And is there a generalization of the nth root of i? Also how would this look graphically on the real number axis? Thanks
0
votes
0answers
25 views

Describe the set of points $0 < |2z - 1| < 2$ on the complex plane

I'm never sure how to do these, minus the tedious algebra. I'm trying to see if I can develop a healthy intuition concerning problems of the kind; I'll summarize my approach here. Let $w = 2z, z \in ...
1
vote
2answers
32 views

Determining the branch of logarithm

I want to determine a branch of logarithm such that $f(z)=L(z^3-2)$ is analytic at $0$. I am not really sure how to find a branch but I will explain few things I tried. Since $z^3-2$ maps $0$ onto ...
3
votes
2answers
51 views

Complex numbers on circle or unit radius

Given three points in the complex plane (i.e. numbers $z_1,z_2,z_3\in\mathbb C$), they define a unique circle (unless they are collinear). When does that circle have radius one? I know how to compute ...
2
votes
2answers
32 views

What is the largest open set $\frac{1}{\cos z-2i}$ is analytic in?

This is a very interesting question that I came across and have never solved any question of this sort. How do I find the largest open set on which $\frac{1}{cosz-2i}$ is analytic? Do I find the set ...
3
votes
1answer
78 views

Cube root of complex number without trigonometric functions

Is there a general equation for a cube root of a complex number that does not exploit De Moivre's Theorem or in any way use trigonometric functions? For example, a square root of a complex number $x$ ...
7
votes
3answers
99 views

Notation For Complex Numbers

I have seen many different notations used for complex numbers. Does it make a difference which notation is used, or is any one notation more standard than another? I see a+bi at ...
0
votes
0answers
40 views

The Real Part of an Imaginary Number

Can any real number be the real part of an imaginary number? Can a mathematical expression which is equivalent to any real number be used as the real part of an imaginary number? If a series ...
1
vote
3answers
35 views

Proof the bijectivity of the exponential function mod $2 \pi i$

I am trying to show that the map $\Psi\ \colon \left\{\begin{align}\mathbb{R}/\mathbb{Z} & \longrightarrow S^1 \\ x & \longmapsto e^{2\pi i x}\end{align}\right.$ is a bijection. I am stuck ...
1
vote
1answer
24 views

Cauchy-Riemann conditions for complex differentiability

The Cauchy-Riemann conditions for the differentiability of $f(z) = f(x + iy) = u(x, y) + iv(x, y)$ in $(x_0, y_0)$ are $$\displaystyle \frac{\partial u (x_0, y_0)}{\partial x} = \frac{\partial v ...
3
votes
3answers
82 views

Describe locus of points $z$ that satisfy $|z+2|+|z-2|=5$

For this problem, as the question says, I am supposed to describe the locus of points $z$ that satisfy the equation: $$|z+2|+|z-2|=5$$ Usually these problems aren't too difficult with a bit of ...
1
vote
2answers
48 views

Complex number problem becomes huge when using formulas - is there any workaround?

Here is an equation I need to solve for $z$ where $z$ is a complex number.(I need to show which complex numbers are solution for this problem): $$\left|\frac{1+z}{1-i\bar z}\right| = 1$$ Here are ...
0
votes
3answers
35 views

How to solve $\left|\frac{1 + a + bi}{1 + b - ai}\right| = 1$

I have a problem with solving following equation: $$\left|\frac{1 + a + bi}{1 + b - ai}\right| = 1$$ (where $a$, $b$ are real numbers and $i$ is an imaginary unit) I tried to simplify its left side ...
4
votes
3answers
134 views

the sum of $1-\frac{1}{5}+\frac{1}{9}-\frac{1}{13}+…$

I thought this was the real part of the series: $\sum_{n=0}^\infty \frac{i^n}{1+2n}$, with $i=\sqrt{-1}$. When taking the real part I am left with: $\sum_{n=0}^\infty \frac{\cos(n\pi/2)}{1+2n}$. I ...
-1
votes
0answers
21 views

Inverse Laplace transform of a particular function

I am interested in evaluating the following Bromwich integral $$\mathcal{I}(t)=\frac{1}{2\pi i}\int^{\gamma+i\infty}_{\gamma-i\infty}\frac{e^{zt}}{1+z^{\beta}}\,dz$$ where $\beta>1$ and ...
0
votes
3answers
44 views

Solutions of the equation $((z-1)/z)^4=1$

The question before this asked to solve $z^4 = 1$ (I found the four roots of 1). I used Euler's formula to solve it and hence solve the question below. I'm just not sure how to go about it exactly. ...
1
vote
1answer
24 views

Checking some work on finding roots

OK, I have the following response function: $$H(\omega) = \frac{1-\omega^2 LC}{1+\omega^2 LC - i \omega RC}$$ I want to find where it becomes $\frac{1}{\sqrt{2}}$. This should be simple enough. ...
0
votes
1answer
36 views

Simplify the following and express in the form $ a + bi$ where $i=\sqrt{-1}$

Write $z = \ln(i)$ as $a+bi$, where $i=\sqrt{-1},\, a,b\in\mathbb R$. Is anybody able to provide me with a hint to lead me in the right direction?
0
votes
2answers
34 views

Simplify the following and express in the form $a + bi$

**(a)**$(1+\sqrt{3}i)^{4i}$ I have no problem finding the solution as I simply convert the expression $1+\sqrt{3}i$ into euler/polar form and arrive at the following: let $z=1+\sqrt{3}i$. Then $$z ...
0
votes
2answers
23 views

find the sketch of y=1 under the mapping $f(z)=z^2$

I am being asked to find and sketch the image of the horizontal line y=1 under the mapping $f(z)=z^2$ This is what I have so far, $u(x,y)+iv(x,y)=f(z)=z^2=(x+iy)^2=x^2 - y^2 + 2ixy$ so we then have ...
0
votes
1answer
24 views

Solution of $p(z)=0$ with $z\in\mathbb C$ and $a_k\in\mathbb R$ for all $k$

Suppose $p(z)=a_0+...+a_nz^n$ with $a_k\in\mathbb R$ for all $k$. How can I prove that if $p(z)=0$ then $p(\bar z)=0$? I know it's true, but how can I prove it?
3
votes
5answers
66 views

How to solve $z^6+i=0$

I'm trying to solve $z^6+i=0$. I would have say that it's equivalent to $$z^6=-i\iff |z|^6e^{i6\arg(z)}=e^{i\frac{3\pi}{2}}\iff|z|^6=e^{i\left(\frac{3\pi}{2}-6\arg(z)\right)}$$ But I'm not able to ...
0
votes
2answers
18 views

Absolute value of a complex number proof

Ok, so I have the following proof. Let $z$ and $w$ be complex numbers. Prove $\lvert z+w \rvert ^2 + \lvert z-w \rvert^2 = 2[\lvert z \rvert^2 + \vert w \rvert^2]$. Using $\vert z \rvert^2=z\bar{z}$, ...