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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5
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0answers
42 views

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. [duplicate]

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. MY ATTEMPT SO FAR: If $f^3$ is holomorphic, then we can ...
0
votes
1answer
37 views

complex number locus

The locus of the complex number Z is a rectangle in the Argand diagram with corners $(-a,0), (a,0), (a,a)$, and $(-a,a)$, where $a>0$. What is the locus of $Z^2$? It could be a relatively easy ...
1
vote
1answer
42 views

How to solve $(-19w + 93\overline w)^4=-1$

How to solve $(-19w + 93\overline w)^4=-1$ , if $w\in \mathbb C$ I really have no direction where to solve this question or at least a hint, can someone help?
2
votes
1answer
53 views

How do I solve $x^4+44=0$ according to de Moivre?

How do I solve $x^4+44=0$ according to de Moivre? I tried to use the formula, but I got roots that are not beautiful numbers. What should the complex roots for this equation be?
6
votes
1answer
196 views

$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...
3
votes
1answer
60 views

solving for z in $|e^z| = 2$

How would I solve for z in the following case: $|e^z| = 2$, now I know that $|e^z| = e^a$ if we let $z = a+bi$ so then equating moduli we get $a = \ln{2}$ But what about $b$? $2 = 2e^{(0+2\pi n)i}$ ...
2
votes
1answer
116 views

Show that $\log \log z$ is analytic

Show that $Log( Log z$) is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0, x ≤ 1$. As of now im not too sure on how to solve this problem, so i was ...
2
votes
1answer
95 views

How to solve $e^{3z} = 1+i$

I am trying to solve $$e^{3z} = 1+i$$ Putting the RHS into modulus argument form, $$1+i = \sqrt{2} e^{(\pi/4 +2\pi n)i}$$ Now what I want to do is equate moduli and arguments, letting $z = a+bi$ ...
6
votes
1answer
199 views

Please help me find a complex number book suitable for me

Its been two weeks since I've joined this site, and I have received wonderful answers to my complex number questions at the shortest time. I am specially very weak in Complex numbers, and I see ...
1
vote
1answer
23 views

How to represent the solution of $z^{2}+2z+5=0$ in in Euler form?

I get the solution of $z^{2}+2z+5=0$ $z=-1+2i \;\; \overline {z}=-1-2i$
1
vote
1answer
55 views

Find an infinite sequence of numbers that can't be the derivatives of power series

First, I noticed that the nth derivative of $f$ at $0$ is $n!\cdot a_n$, but this does not really help me to construct a sequence of numbers that can not be generated by the sequence ...
1
vote
1answer
166 views

raising a complex number to a high power.

we should decide whether the following claims are right or not, and explain our decision. let $w_1,w_2,w_3$ be three different roots for the equation $z^3=1$ a) $w_1^{1991} + w_2^{1991} + ...
1
vote
2answers
52 views

Modulus of z^-3?

What is the result of $|z^{-3}|$ and how can one show it? I know $z = e^{i\omega T}=cos(\omega T) + i\sin(\omega T)$, but I cant go further... I would be glad if someone can explain further.
0
votes
3answers
54 views

Given $z_1, z_2$ prove that $4z^2_1+9z^2_2 = 0$

I need to show that given $z_1 = 9 + 9i$ and $z_2=6-6i$, $$4z_1^2+9z_2^2=0.$$ $$z_1 = 12.7(cos 45 + i sin 45)$$ $$z_2 = 8.5(cos 315 + i sin 315)$$ I changed the terms to polar form, applied De ...
0
votes
1answer
23 views

Solve ${z_1/\overline{z_2}} = z^3$

$$z_1= 4\sqrt{2}-i4\sqrt{2}$$ $$z_2= \cos{135^\circ} +i\sin{135^\circ}$$ Find all the complex numbers $z$ that fulfill the following equation: $${z_1\over \overline{z_2}} = z^3$$ be aware that ...
0
votes
1answer
52 views

How to sketch the following set?

Sketch on the argand's diagram the following set: $A=\{w \in \mathbb{C} \backslash \{0\}:w^3-w^{-3} \in \mathbb{R}\}$. How to approach this question?
0
votes
1answer
34 views

Sum of modulus of complex numbers

I'm trying to establish if |sin(z)|+|cos(z)| is greater than or equal to 1. I have tried to write out the expression in exponential form, but I don't really arrive at anything useful. I would really ...
3
votes
4answers
93 views

Conceptual question about the imaginary number $i$

One of the first things we see in our first complex analysis class is the standard way of introducing us to the imaginary unit $i$ which is to think of a solution to the equation $$x^2=-1$$ Obviously, ...
1
vote
4answers
228 views

How to find the roots of $x^4-i=0$

I need the manually analysis to calculate the roots without using the numerical methods
0
votes
1answer
77 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
0
votes
2answers
107 views

Show that complex numbers are vertices of equilateral triangle

1)Show if $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$ then $z_1,z_2,z_3$ are vertices of equilateral triangle inscribed in a circle of radius. I thought I can take use from roots of unity here, since ...
1
vote
1answer
170 views

Best way to solve $ z^{2} + i \cdot z = 0 $

What is the best way to solve $ z^{2} + i \cdot z = 0 $ ? I have tried to solve it via completing the square and using the quadratic formula, and got different answers.
2
votes
3answers
74 views

Show complex solutions exist

Let A be a complex number and B a real number. Show that the equation $\,\lvert z^2\rvert+ \mathrm{Re}\, (Az) + B = 0\,$ has a solution iff $\,\lvert A^2\rvert \geq 4B$. If this is so, show that the ...
0
votes
3answers
32 views

Are conditions equaivalent that they are roots of unity?

If I have conditions that $|z_1|=|z_2|=|z_3|=|z_4|=1$ and $z_1+z_2+z_3+z_4=0$ Is it suffice to state they are roots of unity ?
1
vote
3answers
54 views

Prove Complex Relationship

Question: Prove that if $z + \frac{1}{z}$ is real then either the magnitude of $z = 1$ or $z$ is real. I'm struggling with, I found that $2ab$ must be equal to $0$ but I don't see how that help. ...
0
votes
1answer
43 views

Proof about field extension : A geometric way

Let $M \subset \mathbb C $ be a sub-field which is not contained in $\mathbb R$ and which is closed under complex-conjugation. Let $L(M)$ be the set of all lines which crosses two points of $M$ and ...
2
votes
3answers
31 views

Show Trigonometric Identities from Complex indentity

So the exercise says to show $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ and $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$ By using the following identity: $e^{i(a+b)}=e^{ia}e^{ib}$ How do ...
-1
votes
3answers
60 views

Why is the following equivalent transformation of the imaginary number legitimate? [duplicate]

Why is this substitution acceptable? $\sqrt{i}=\frac{1+i}{\sqrt{2}}$
3
votes
1answer
114 views

Riemann Zeta Function at $s = 1 + 2 \pi i n / \ln 2$

I am aware that the function defined by $$ \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots $$ for $Re(s)>1$ can be extended to a function defined for $Re(s)>0$ by writing $$ ...
2
votes
2answers
31 views

Find the Power Series

How would one write $f(z) = \frac{1}{1-wz}$ as a power series? ( Where $z,w$ are in $C$.) Would it just be $\sum_{n=0}^{\infty} (zw)^n$?
7
votes
1answer
126 views

Inequality relating coefficients and roots of a complex polynomial

While going through some olympiad handouts I stumbled upon a problem related to an upper bound for the Mahler measure, which stated that Given a polynomial $f(x) = x^n + a_{n-1}x^{n-1} + \dots + a_0 ...
0
votes
2answers
127 views

complex numbers

I have a number of questions about complex numbers and I need your help: $z_1, z_2, z_3, z_4, z_5$ are complex numbers that fulfil |z1|=|z2|=|z3|=|z4|=|z5|=1 prove that $|z_1+z_1+z_3+z_4+z_5| = ...
0
votes
2answers
39 views

complex numbers two problems

1) If $z=\cos\alpha+i\sin\alpha$ for $\alpha \in[0, 2\pi]$ then find $\alpha$ for $z^2+z$ I transform to this moment $\displaystyle ...
0
votes
2answers
59 views

Why is $t=\frac{1}{2}$ a root for $\tan 4\theta= \frac{4t-4t^3}{1-6t^2+t^4}=\frac{-24}{7}$, where $t=\tan \theta$

Show that $(2+i)^4=-7+24i$ $$\cos 4\theta = \cos^4 \theta - 6\cos^2 \theta \sin^2 \theta + \sin^4 \theta$$ $$\sin 4\theta= 4\sin \theta \cos^3 \theta- 4 \sin^3 \theta \cos \theta$$ ...
0
votes
1answer
25 views

z and w are two complex numbers prove the relationship

If $z$ and $w$ are complex numbers such that $|z+w| = |z-w|$ Prove that $\arg z - \arg w = \pm \ \pi/2$ Can someone please help me?
0
votes
1answer
309 views

finding equation of circle in complex plane

So i was asked to find the equation of the circle going through 1, i, and 0 Now i know that the equation of circle in complex form is: | z - (Zo) | = r where r is radius. So based on these values, ...
2
votes
1answer
119 views

How to find the roots of $(\frac{z-1}{z})^5=1$

Write down the fifth roots of unity in the form $\cos \theta + i \sin \theta$ where $ 0 \leq \theta \leq 2\pi$ Hence, or otherwise, find the fifth roots of i in a similar form By writing ...
1
vote
1answer
115 views

Describing a subset of the complex plane formed by z satisfying |z-i| + |z+i| = 3

I have been asked to describe the subset of the complex plane which is formed by the complex numbers z satisfying |z-i| + |z+i| = 3. It was easy to see that if the points z lie on the line segment ...
1
vote
1answer
288 views

Proving the reverse triangle inequality of the complex numbers

I'm having trouble understanding this proof for the reverse triangle inequality of the complex numbers. Suppose for any $z, w \in \mathbb{C}$, we have $|z + w| \leq |z| + |w|$ (the triangle ...
0
votes
3answers
83 views

How to compute $(i^2-i^4+i^6-i^8+…+i^{38})^2$

How can i compute $(i^2-i^4+i^6-i^8+...+i^{38})^2$ ? I can see that the powers are arithmetic progression with $d=2$ but i tried to compute $S_{19}$ but it didn't work. Thanks.
1
vote
3answers
46 views

what is $c^{a+\mathrm i b}$ for $c \in \mathbb{R}$?

How can be $c^{a+\mathrm i b}$ for $c,a,b \in \mathbb{R}$ rewritten in the form $e+ \mathrm i d$ for $d,e \in \mathbb{R}$ (i.e. as a $\mathbb{R}$-linear combination of $1, \mathrm i$)?
2
votes
2answers
48 views

maximum, complex quadratic function, Is my solutions correct?

I'm trying to compute $\max_{|z| \le 1} |(z+2)(z-1)|$. Here's how I do it: $\{z \in \mathbb{C} \ | \ |z| \le 1 \}$ is compact and $f(z) = (z+2)(z-1)$ is continuous, so it suffices to look for ...
2
votes
2answers
121 views

How to find the roots of $(w−1)^4 +(w−1)^3 +(w−1)^2 +w=0$

Write down, in any form, all the roots of the equation $z^5 − 1 = 0$ Hence find all the roots of the equation $$(w−1)^4 +(w−1)^3 +(w−1)^2 +w=0$$ and deduce that none of them is real ...
2
votes
5answers
243 views

Why non-real means only the square root of negative?

Once in 1150 AD, an Indian mathematician Bhaskara wrote in his work Bijaganita (algebra) that, There is no square root of a negative quantity, for it is not a square However later on in 1545 an ...
2
votes
2answers
40 views

Negative imaginary exponents

I was reading this question earlier: Understanding imaginary exponents In the answer, the answerer says $A^i=x+iy$ Furthermore, we can write $A^{−i}=x−iy$ for the same $x$ and $y$. Can ...
7
votes
1answer
65 views

How do I find a constant for a polynomial so its roots are reflective around a linear function?

How can I find all complex numbers $w$ so that the roots of the following polynomial are reflected around a linear function $f(x)$ $$p(q) = q^2-4q+w = 0$$ If I want to find all the complex numbers ...
1
vote
1answer
62 views

Zeroes of the floor (greatest integer function)

I was just browsing Wolfram Alpha and looking at interesting graphs of functions when I noticed that it gave complex numbers for the numerical roots of $\lfloor x\rfloor$. Does anybody know why?
0
votes
1answer
15 views

complex number conjugates (simple)

Show, by squaring both sides, that $|z - 10i| = 2|z-4i|$ is equal to $zz^* - 2iz^* + 2iz -12 = 0$ The bit I'm really stuck on (reading through the answers) is how $(z-10i)^2 $ is equal to $(z - ...
1
vote
1answer
49 views

Third degree polynomial with unknown coefficients $q^3-3aq^2+b^2q+c = 0$

For an equation $q^3-3aq^2+b^2q+c = 0$ we know the roots $c, (a+b), (a-b)$. What is a good place to start with such equations? I've tried setting up a system of equations, but this is supposed to be ...
0
votes
2answers
57 views

Real part of a complex number divided: $\Re\frac{z+1}{z-1}=0$

$\Re\frac{z+1}{z-1}=0$ I've tried so many methods, they all end up with two variables $a, b$. I tried setting $z= a+ib$. This give me the equality $2\cdot\Re\frac{z+1}{z-1} = ...