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
34 views

Taylor series on complex analysis

Suppose that, I have $\sum_{n=1}^\infty (z^n)/n$. Now clearly for the open disk $|z|<1$, above series converges. But if I consider $|z|=1$, then clearly for $z=1$, above series diverges. How do I ...
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1answer
27 views

Creating a Hermitian matrix that is also positive semi-definite

Given some measurements on empirical data (in the form of a multigraph with two weighted edges between every pair of vertices), I would like to place the measurements in a Hermitian matrix that also (...
2
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1answer
71 views

Are there any solution for a,b,c,d such that $(a+bi)^{n}+(c+di)^{n}=2i$

Are there any solution for a,b,c,d such that $(a+bi)^{n}+(c+di)^{n}=2i$. With a,b,c,d,n are positive integer numbers and $a+bi, c+di$ are complex numbers . I just have started learning about comlex ...
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1answer
34 views

Show that a set of polynomials are linearly independent in the complex space

I have been trying the solve the following question without any success: Let $\lambda_1, \lambda_2, \lambda_3$ be three distinct complex numbers and define the polynomials $m(\lambda), m_1(\lambda), ...
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1answer
37 views

How do you show that f(z)=z conjugate isn't linear?

let $x_1= a+ib,x_2= c+id,k=$scalar $f(x_1,x_2)=f(x_1) + f(x_2)$ $f(a+ib + c + id)=(a+c)-i(b+d)$ $f(a+ib)+f(c+id)=(a+c) - i(b+d)$ $f(kx_1)=kf(x_1)$ $f(k(a+ib))= k(a-ib)$ $kf(x_1)=k(a-ib)$ Looks ...
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1answer
40 views

Powers of complex numbers

Prove that $\left(\sqrt{3}-i\right)^n = 2^n \left(\cos(n\pi/6)-\sin(n\pi/6)\right)$ $(1+\cos\alpha+i\sin\alpha)^n = 2^n\cos^n(\alpha/2)(\cos(n\alpha/2)+i\sin(n\alpha/2))$ I am completely lost with ...
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1answer
32 views

Complex conjugate of $z$ as a different variable

Can a complex conjugate be represented by a different letter than $z$? As in: Let $y$ be a complex number satisfying $|y|<1$. Find the set of all complex numbers $z$ satisfying $|z-y|\le|1-\bar{...
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2answers
84 views

(complex analysis) Prove that: $\arg ((z_3-z_2)/(z_3-z_1)) = 1/2 \arg z_2/z_1$

If $|z_1|=|z_2|=|z_3|$ Urgent help needed. I have used: $z_1=x_1+\mathrm iy_1,z_2=x_2+\mathrm iy_2,z_3=x_3+\mathrm iy_3$ and obtained $$\arg\frac{z_3-z_2}{z_3-z_1} = \arctan \frac{(y_3-y_2)(x_3-x_1)-(...
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1answer
50 views

Rewriting a trig function into a sum of exponential functions

Rewrite the function $2 + 4\sin(\pi t + \frac{\pi}{6})$ into a sum of exponential functions. By that I mean using Euler's formula $\sin(x) = \dfrac{e^{i\pi x} - e^{-i\pi x}}{2i}$. If it wasn't for ...
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2answers
49 views

Find the argument of $\dfrac{(3-2i)(1-i)}{(2+i)^2}$

As the header suggests, I am supposed to find the argument for the complex number $\dfrac{(3-2i)(1-i)}{(2+i)^2}$ This is how I've tried: Approach 1: Calculate the arguments by factoring out the ...
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1answer
37 views

Find the analytic function

$f(z)=1 $ satisfies the condition Using Identity Theorem $f(z)=1$ can be only function that satisfies this. so option (b) is NOT true. Am I on correct path?
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1answer
30 views

On charge conjugation of Dirac spinor

Suppose we have Weyl spinor $\psi_{a}$, which transforms under irreducible representation $\left( \frac{1}{2}, 0\right)$ of the Lorentz group, $$ \psi_{a} \to (T(g))_{a}^{\ b}\psi_{b}, $$and complex ...
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1answer
49 views

why $f(z) = z^{(3/2)}$ does not have derivative at z = 0 in complex plane.

it seems that the $f'(z) = z^{(1/2)}$ means that this function has derivative for every complex value. But why $f(z) = z^{(3/2)}$ does not have derivative at z = 0
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1answer
169 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 ...
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0answers
27 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 ...
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1answer
48 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
103 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
42 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 $...
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2answers
74 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 z=\frac{1}{2}(e^...
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1answer
96 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 ...
2
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1answer
33 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
52 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 conjugation?)...
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2answers
25 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
44 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
79 views

Periodic function without trigonometry and complex numbers [closed]

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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2answers
54 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 $\bf{My\; Try::}$ Let $...
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1answer
67 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 ...
2
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1answer
149 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
46 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 = ...
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1answer
134 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
41 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
228 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
48 views

Doubt in raising a power to a complex number [duplicate]

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
26 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 ...
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1answer
112 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 ...
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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
29 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
100 views

Find the number of elements of a complex subset [duplicate]

How many elements does the set $\{z\in \mathbb C:z^{60}=-1,z^k\not=-1\text{ for } 0<k<60\}$ have ? (A) $24$ (B) $30$ (C) $32$ (D) $45$. Which is correct ? $z^{60}=-1=\cos(2k\pi+\pi)+...
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1answer
34 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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1answer
39 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
36 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
39 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 ...
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votes
4answers
158 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
44 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 end....
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0answers
38 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
99 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
79 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
49 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: $$x^2=1=e^{(0i)}$$ ...
0
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2answers
54 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 ...
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1answer
74 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?