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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1answer
28 views

Prove that $\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$

As the title suggests, what is required to prove is that $$\tan5 \theta = \frac {5\tan \theta -10 \tan ^3 \theta +\tan ^5 \theta} {1-10\tan ^2 \theta +5\tan ^4 \theta}$$ I was looking back through my ...
0
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0answers
7 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
1
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1answer
15 views

There exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$. Show that $f$ can be extended analytic on $\Bbb C$.

(a) Suppose that $f$ is analytic on the open unit disk $\{z: |z|<1 \}$ and there exists a $M$ such that $\mid f^k (0)\mid \leq k^4 M^k$ for all $k \geq 0$. Show that $f$ can be extended analytic on ...
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2answers
62 views

Why is Euler's formula a definition?

Even though there are proofs for Euler's formula for complex exponentials (see wikipedia for instance), it is mentioned as a "definition" in most textbooks. Why is that? My understanding is that a ...
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3answers
27 views

Let $A$ be a complex number and $B$ be a real number. Prove that $\mid z^2\mid+Re(Az)+B=0$ can only have a solution iff $\mid A^2 \mid \ge 4B$.

Been stumped on this question for a while. I tried letting $z=\mid z \mid \cdot e^{i \alpha}$ and $A=\mid A \mid \cdot e^{i\beta}$ -- assuming that $\alpha$ and $\beta$ were the arguments of $z$ and ...
3
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2answers
37 views

Does the identity ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ given in my text hold?

In my text book I saw that ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ But when I tried deriving it myself I got this: $${|\cosh z|}^2={\cos}^2y+{\sinh}^2x$$ See my working below: $$\cosh ...
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1answer
25 views

Find the location of the center and the radius of the following circle: [on hold]

Find the location of the center and the radius of the following circle: $$ \left| \ \frac{z-1}{z+1} \ \right| \ = \ 3 \ \ . $$ $ \ z \ $ is a complex number. Thanks in advance!
3
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0answers
46 views

Find a convergent solution for $a$

Find a value for $a$ in which the following sum converges. $$a+a!+(a!)!+((a!)!)!+\cdots$$ I know that there are no solutions if you only look at $a\in \Bbb{R}$, but are there any solutions if you ...
0
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0answers
28 views

How can we represent complex numbers in 2-d plane(i.e. complex plane) if there is no ordinal relationship between them? [on hold]

If there is no ordinal relationship(2i is not greater or equal or lesser than i i.e there is no order relation between i and 2i) in complex numbers then why are they represented in ordinal manner in ...
2
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1answer
20 views

A set $I$ of isolated complex numbers such that $[0,1]\subset\{Re(z):z\in I\}$

Is there a set $I$ of isolated complex numbers, such that $$[0,1]\subset\{Re(z):z\in I\},$$ where $Re(z)$ is the real part of the complex number $z$.
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0answers
21 views

about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$

I am a little confused about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$. From other answers (Is a complex vector space closed under complex ...
1
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2answers
19 views

complex no. $z$ such that $ |z|<\frac{1}{3}$ and $ \sum^{n}_{r=1}a_{r}z^{r} = 1\;,$ Where $|a_{r}|<2$

Prove that there exists no complex no. $z$ such that $\displaystyle |z|<\frac{1}{3}$ and $\displaystyle \sum^{n}_{r=1}a_{r}z^{r} = 1\;,$ Where $|a_{r}|<2$ $\bf{My\; Try::}$ We can write ...
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1answer
31 views

Having trouble solving a problem involving hyperbolic trignometric functions

We have to find the value of $$ \tanh^{2}a * \cosh^{2} b - \cos^ {2} c \, $$ if $$\sin(a+ib) * \sin(c+id) = 1.$$ Can anyone solve this? Pls share the solution
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2answers
62 views

Periodic function without trigonometry and complex numbers [on hold]

Can I get a periodic function without using trigonometric functions or complex numbers? UPDATE: The question has been superseded by Single-statement Continuous Periodic function without trigonometry ...
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0answers
29 views

Exercise about factorization

I've just started a new year at school, and I learned these formulas: $\sin x = \frac{e^{ix} - e^{-ix}}{2i}$ and $\cos x = \frac{e^{ix} + e^{-ix}}{2}$ We used them in class to do some factorization ...
2
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2answers
24 views

If complex no. ($z$) satisfying $\frac{1}{2}\leq |z|\leq 4\;,$ Then Max. and Min. of $\left|z+\frac{1}{z}\right|$

Let $z$ be a complex no. satisfying $\displaystyle \frac{1}{2}\leq |z|\leq 4\;,$ then the Sum of greatest and least value of $\displaystyle \left|z+\frac{1}{z}\right|$ is $\bf{My\; Try::}$ ...
0
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1answer
48 views

Resolve $A=\cos{(\pi/7)}+\cos{(3\pi/7)}+\cos{(5\pi/7)}$ using $u=A+iB$

With these two sums: $$A=\cos(\pi/7)+\cos(3\pi/7)+\cos(5\pi/7)$$ $$B=\sin(\pi/7)+\sin(3\pi/7)+\sin(5\pi/7)$$ How to find the explicit value of $A$ using: $u=A+iB$ the sum of $n$ terms in a ...
1
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1answer
30 views

Inequality for the gradient of a power of absolute value

Let $U \subset \mathbb{R}^2$ be open, and let $f : U \to \mathbb{C}$ be a smooth complex-valued function which does not vanish anywhere on $U$. Let $r > 0$ be a real constant. Does the ...
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1answer
12 views

Show that a line is tangent to a circle in the extended complex plane.

The straight line $l$ in the extended-complex plane pasess through $2+i,2+2i$.The circle $C$ centered at $-1-2i$ with radius $3$. First, I find the parametrization of the straight line which is $$z = ...
3
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1answer
122 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me ...
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0answers
38 views

Is the square root of -1 really “i” [duplicate]

I know that the imaginary unit i is a number with the following property: i^2 = -1 But I often see people turn that into this ...
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2answers
107 views

Proving that a complex number lies on the imaginary axis.

Given that there are two complex numbers - $z, w$ - such that $w\overline{w} = 1$ and $z = \frac{1+w}{1-w}$, how do I deduce that $z$ lies on the imaginary axis?
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1answer
36 views

Doubt in raising a power to a complex number

What's the value of $$i^{i^{i^{...}}}$$? I tried to take log on both sides. $x=i^x$ $\implies \log x=x \log i$ After this how can I solve this... I am sorry, that I don't know the methods you ...
2
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1answer
21 views

Let $z,w\in \mathbb{C}$ where $|w|<1$ (modulus). What is the set of all $z\in \mathbb{C}$ that satisfies $|z-w|\leq|1-\bar{w}z|$.

Let $z,w\in \mathbb{C}$ where $|w|<1$ (modulus). What is the set of all $z\in \mathbb{C}$ that satisfies $|z-w|\leq|1-\bar{w}z|$. I've tried a few things with no luck. I wrote $z,w$ are complex ...
1
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1answer
38 views

Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...
1
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1answer
36 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
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2answers
25 views

calculating complex numbers - help needed [closed]

Probably it is simple, but I am blind right now and I do not see how to solve this task: $e^{i \frac{2\pi}{3}}+e^{i\frac{4\pi}{3}}+1$
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1answer
34 views

Find the number of elements of a complex subset

How many elements does the set $\{z\in \mathbb C:z^{60}=-1,z^k\not=-1\text{ for } 0<k<60\}$ have ? $z^{60}=-1=\cos(2k\pi+\pi)+i\sin(2k\pi+\pi)$. Then , ...
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1answer
23 views

Expression of reflection isometry in the complex plane

Using the fact that an anti-displacement in the plan has the form $$f(z) = a \overline{z} + b$$ I have done some computation to find the reflection about the line passing through two points $P$ and ...
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3answers
82 views

Find all complex numbers satisfying $x^4+x^2+1=0$ [closed]

Find all solutions that fit: $$x^4+x^2+1=0$$ I did it couple of days ago, now I can't remember.
1
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1answer
18 views

Solve complex equation graphically

I have this problem that is split in 2, A and B, and Im struggling with B in particular, but I also dont know if I have done A correctly, which I suppose is necessary.. A) "Let $z$ be the complex ...
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0answers
32 views

Fourier series for a Sinusoid in a conventional way?

So my TA in class introduced this amazing way of finding fourier series coefficients for a sin wave, by writing $ sin( \omega t ) = (e^{i\omega t}-e^{-i\omega t}) / 2i $ ----(1) Hence getting the ...
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2answers
22 views

Solve Complex Equation with several terms

I have a complex number $z = 3 + 3i$ And I want to find all solutions of $z^{10} + 2z^{5} + 2 = 0$ I'm kinda lost. I recognise the fact that I can substitute $u = z^{5}$ and rewrite the equation as ...
2
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4answers
112 views

Is $i^i$ mathematically valid? [duplicate]

WARNING: SLIGHT NSFW http://www.smbc-comics.com/index.php?db=comics&id=2934#comic Uhh...guys, mathematically speaking, how accurate is this comic. From what I remember in High School $$a^b= ...
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1answer
37 views

Exponential Complex Number

I need assistance in solving the following: http://i.stack.imgur.com/EcGLD.jpg I am not very sure on how to remove the exponential to convert it into complex numbers and get the arguments in the ...
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0answers
28 views

Is there a name for complex numbers over affinely extended reals?

Is there a name for the set of complex numbers over affinely extended real line, that is $\mathbb{C}\cup \{-\infty\}\cup\{+\infty\}$? I think this set is the most commonly used in analysis ...
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3answers
38 views

$|z|=1$ should represent semi-circle or circle?

Suppose we have complex number $z=x+iy$ and we are given locus $|z|=1$ which should be $\sqrt{x^2+y^2} =1$ this should be a semi-circle above x axis , it's when we square our equation we get a circle ...
2
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1answer
40 views

Find the min and max distance from origin of the curve $\vert z+\frac{1}{z}\vert=a$

$z$ is a complex number, by the way. I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem. One of ...
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2answers
38 views

multiple sets of complex roots of a number?

I am not sure if this question was asked before but I couldn't find the right keywords to choose for searching. So today I discovered a weird problem: If we take this equation: $$x^2=1=e^{(0i)}$$ ...
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2answers
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
1
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1answer
35 views

Find the mistake in calculation [duplicate]

$(-1)^3 = (-1)^{6/2} = ((-1)^6)^{1/2} = 1^{1/2} = 1$ So it comes $(-1)^3 = 1$ can anybody explain where exactly the mistake in calculation?
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2answers
25 views

the locus of $z$ in the complex plane

Describe the locus of $z$ in the complex plane if $z$ satisfies: $$ arg(z)=arg(z+3+i)\quad (mod\ 2\pi) $$ Indeed Let $O$ be the origin and $B=-3-i$. \begin{align*} arg(z)&=arg(z+3+i)\quad ...
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3answers
61 views

Finding the modulus and argument of a complex number

I would need help with this question: $$Z = \frac{(1+j2)^2(4-j3)^3 }{ (3+j4)^4 (2-j3)}$$ My starting point for this question is to expand the complex numbers first then continue doing but after ...
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0answers
21 views

Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, then $u_n \rightarrow 0$ uniformly on compact subsets of $D$.

Let $D \subseteq \Bbb C$ be a connected open subset and let {$u_n$} be a sequence of harmonic functions $u_n: D \rightarrow (0,\infty)$. Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, ...
2
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2answers
46 views

What is the interval for possible values of the argument of a complex number?

It looks like there are different intervals in which the argument of a complex number can be. Some say it goes from $-\pi$ to $+\pi$ others say it goes from $0$ to $2\pi$. For the most part, both ...
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0answers
17 views

If $f:B_N\rightarrow \mathbb{D}$ and $z_n\in B_N$ with $\{f(z_n)\}$ thin, is $\{f(\phi(z_n))\}$ thin for any autmorphism $\phi$ of $B_N$?

Let $B_N$ denote the open unit ball in $\mathbb{C}_N$. A sequence $\{z_n\}$ of distinct points in $\mathbb{D}$ is called thin if $\lim_{k\rightarrow \infty}\displaystyle\prod_{j: j\not =k}^\infty ...
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2answers
43 views

Reducing to upper triangular form

I've just had some difficulty with this transforming this matrix into upper triangular form: $$ \pmatrix{ i& 2i& -1\\1 & 1& i\\ 2-i& 1& i } $$ I've tried almost everything. ...
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2answers
37 views

Express the complex number in rectangular form $a + ib$

$12e^{2-\pi*i/3}$ express this in rectangular form $a + i\cdot b$ Not sure how to solve when fractions are involved Example $2.6\cdot e^{3+i} = 2.6\cdot e^3\cdot e^i$ ?
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2answers
67 views

2+2=square root of 16. What's the appropriate answer? [closed]

4? Positive and negative 4? I just got into an argument with a buddy about this. He argues if it's not an i, it's not included as a imaginary number, but only the real positive number.
3
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3answers
275 views

Finding complex number defined by 3 equations

Let $z$ be a complex number satisfying $$\DeclareMathOperator{\Re}{Re}\Re[z^4]=1/2$$ $$z\bar{z}+2|z|-3=0$$ $$\arg z \leq \frac{\pi}{4}.$$