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Questions tagged [complex-numbers]

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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9 views

Fourier analysis, complex conjugate and orthogonality

To obtain the magnitude of a particular frequency contained in a periodic signal, you take the inner product of the signal’s function with the (analyzing) basis function corresponding to such ...
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2answers
33 views

To prove $\tan \phi_m + \sec \phi_m =(\tan \phi_1 + \sec \phi_1)^m $

If $\phi_1, \phi_2, ... $ is a series of positive acute angles so that $\tan \phi_{m+1} = \tan \phi_m \sec \phi_1 + \sec \phi_m \tan \phi_1$ then prove that- $$\tan \phi_{m+n} = \tan \phi_m \sec \...
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0answers
36 views

$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})…(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ . Find $P$.

$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is the nth root of unity. Find ...
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0answers
13 views

Using rules of spiral enlargement.

If I may summarize the steps used in this simplification: They express the spiral enlargement in terms of a polar complex number. They multiply this result with $\vec{AT}$ to get $\vec{AW}$. To find ...
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0answers
14 views

What is the range of koebe function? [on hold]

I don't know about the behaviour of koebe function k(z)=z/(1-z)^2.
1
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1answer
49 views

Show that $\oint_{\gamma} \frac{Log z}{z}dz=0$

I need to prove that $$\oint_{\gamma}\frac{\log z}{z}dz=0$$ where $\log z= \log |z|+ \text{Arg } z$ ($\text {Arg}$ is the principal argument), $\gamma:z=e^{it},\quad 0\leq t\leq 2\pi$. I know ...
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0answers
23 views

Analytics function satisfied certain equality

Question: If $f(z)$ is analytic in a domain $D$, $f'(z)\neq 0$, $|f(z)|\neq 1$, then show that $$w=log\frac{|f'(z)|}{1-|f(z)|^2}$$ satisfy $\nabla^2w=4e^{2w}$. I don't know how to start with this ...
2
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4answers
174 views

Why is $e^{2πi} = e^0$ true while $2πi = 0$ is false?

I came across a perplexing thing while testing some assumptions in Wolfram|Alpha, and as I don't have a math education beyond college algebra, I thought this would be a good place to ask. I would just ...
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3answers
90 views

Is $i = e^{\frac{\pi}{2}e^{\frac{\pi}{2}^{.^{.^.}}}}$? [on hold]

I came up with this and I am wondering if it is true, because it seems illogical that $i$ can be made from an infinite power tower of reals. The way I found this is the following: $$i=e^{\frac{\pi}{2}...
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2answers
17 views

Show that $2\cos(mp-nq)$ is one of the values of $\left( \frac{x^m}{y^n}+\frac{y^n}{x^m} \right)$

Q:If $2\cos p=x+\frac{1}{x}$ and $2\cos q=y+\frac{1}{y}$ then show that $2\cos(mp-nq)$ is one of the values of $\left( \frac{x^m}{y^n}+\frac{y^n}{x^m} \right)$My Approach:$2\cos p=x+\frac{1}{x}\...
6
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1answer
918 views

A visually guided proof of the fundamental theorem of algebra?

A complex root of a polynomial $P(z)$ is a pair of real numbers $u,v$ that simultaneously make the real part and the imaginary part of $P(z)$ zero. The zeros of the real part and the imaginary part ...
3
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2answers
70 views

Explicit formulas for $\operatorname{Re}(z^n)$ and $\operatorname{Im}(z^n)$

I'm looking for a closed formula for the real and imaginary part of $z^n = (u + iv)^n$. We have $$\operatorname{Re}(z^{n+1}) = u\operatorname{Re}(z^{n}) - v\operatorname{Im}(z^{n})$$ $$\...
1
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1answer
33 views

Can we define an order such that all complex numbers are equal?

There is an exercise "Prove that no order can be defined in the complex field that turns it into a field." in baby Rudin's book. And I can get a contradiction if $i>0$ or $i<0$. However, I ...
1
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4answers
66 views

Prove that $\frac{1+\sin\theta + i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta.$

Prove that $$\frac{1+\sin\theta + i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta.$$ Hence, show that $$(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5})^5+i(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{...
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2answers
38 views

How can we prove that 4 is a complex number?

We know 4 is a real number but how can we prove that it is a complex number? How can be describe it in the a+ib form??
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0answers
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0answers
16 views

Proving that a conformal mapping preserves the real analyticity of function

I'm reading the lecture notes for my one of my physics courses on scattering theory and in it, I came across a statement that I really don't quite get. Basically we start with a function $f$ that's ...
1
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1answer
36 views

Proving if $z$ is an n'th root, $\bar z$ is also an n'th root

Let $n>0$ be an even number, and let $z$ be an $n$'th root of a real number. Is $\bar z$ also an $n$'th root of this number? My answer is yes. The way I solved this was to consider a complex ...
1
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0answers
33 views

If $P(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+…+a_1z+a_o$ a polynomial whose coefficients are all real. [duplicate]

If $P(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+...+a_1z+a_o$ a polynomial who's coefficients are all real. Explain why if $z_1$ is a root of $P$, then $\bar{z_1}$ is also a root. And give an example as ...
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2answers
29 views

How to find the rational root in this case. Is there a relationship between all the roots? [on hold]

I know that the complex roots come in pairs and I know the Vieta's formulas. But I still don't know how to deal with this problem.
1
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3answers
303 views

Complex numbers proof with modulus argument question

I'm having trouble with this complex numbers proof. Prove $|(z-a)/(\overline az-1)|=1$ if $a,z$ are any complex numbers, where $z\ne a$ and $|z|=1$. I tried substituing $z,a$ for general Cartesian ...
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1answer
28 views

Given that $z = (1+ bi) ^{2}$, find the exact value of b if $arg z = \frac{\pi}{3}$ and b is real and positive.

Essentially the title. This is a homework problem, and I tried to do it by considering it as a $1 ,\sqrt{3},2$ triangle as the angle is $\frac{\pi}{3}$, and thus $b$ would be $\sqrt{3}$. However, ...
0
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1answer
30 views

Is it true that if the complex function $\bar{f}$ is differentiable, then $f$ is differentiable?

If $\bar{f}=u(x,y)-iv(x,y)$ is differentiable, then the first-order partial derivatives of $u$ and $-v$ exist and they are continuous. Also, the Cauchy-Riemann equations hold, i.e., $u_x=-v_y$ and $...
1
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1answer
24 views

Complex Recursion

Consider the recursive function $C_n = C_{n-1} + iC_{n-2}$, where $C_1 = 1, C_2 = 1.$ If $C_{10}$ is written in the form $a+bi,$ find $b$. I solved this problem through brute force with a ...
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1answer
47 views

How to solve $2z^5 + z^4 -6z^2 + z + 1 = 0$?

$$2z^5 + z^4 -6z^2 + z + 1 = 0$$ z is complex number. I tried to make factor but i didn't find.
2
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1answer
59 views

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and $f$ is differentiable at z.

Show $\frac{df}{dz}=\frac{\hat{r}\bullet\nabla f}{e^{j\phi}}$ . Where $z$ is a complex number and f is differentiable at $z$. The $\bullet$ denotes the dot(inner) product. $\nabla$ is the gradient. $...
0
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3answers
27 views

If |f(z)-w|<|w|/2 how do you conclude |f(z)|>|w|/2? z and non-zero w are complex numbers.

If |f(z)-w|<|w|/2 how do you conclude |f(z)|>|w|/2? z and non-zero w are complex numbers. f is a function of z. The inequality is defined on a domain S in complex plane.
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1answer
17 views

Prove that $arg(z') = \frac{\pi}{2}+(\vec {BM}\;;\vec{AM}) +2k\pi$ where k is an integer.

Consider the points M of affix $z$ $(z \neq 2i)$ , and $M'$ of affix $z'$ such that $z'= \frac{i(z-2)}{z-2i}$ . A and B are the points of respective affixes 2 and 2i. Show that $arg(z') = \frac{\pi}{...
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0answers
16 views

can I use the following formula for polar coordinates as well?

Simpl asking, can I use the following formula for $\theta$ as we always do for an argument of a complex number $z=a+bi~$: $$\theta=\tan^{-1}(b/a),~~a>0;~~~ \theta=\pi+\tan^{-1}(b/a),~~a<0$$ ...
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3answers
39 views

$\int^0_{-\pi} {i e^{2it}\sin(2t)}$ by parts?

By integrating by parts the following: $$\int^0_{-\pi} {i e^{2it}\sin(2t)}dt$$ with $$u=\sin(2t); v'=ie^{2it}$$ $$u'=2\cos(2t);v=\frac{1}{2}e^{2it}$$ I get: $$\int^0_{-\pi} {ie^{2it}\sin(2t)}dt = ...
1
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1answer
24 views

Using the fifth root of unity to show the cosine equation

Consider the equation expressed the fifth root of unity: $z^5-1=0$ To show that: $$2(\cos(\frac{2\pi}{5})+\cos(\frac{4\pi}{5}))=-1\\4\cos(\frac{2\pi}{5})\cos(\frac{4\pi}{5})=-1$$ I have already ...
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0answers
24 views

Generalizations of the field concept to triplets of elements

In the 'Field' article at MathWorld there is a phrase that I don't understand: It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements ...
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0answers
27 views

solve problem $|e^{iz}|$ [on hold]

Please help me to solve $|e^{iz}|$?
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1answer
53 views

Doesn't $i=-i$ from the definition $i^2=-1$? [duplicate]

There is 2 definition about imaginary numbers:- Imaginary unit, $i=\sqrt{-1}$ The square of Imaginary unit, $i^2=-1$ But the later is used mostly, as known to me, because of many reasons. And my ...
2
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1answer
48 views

Let $f$ be the disk $D[0, 1]$ be a holomorphic function. Show the following

I was able to prove a) using the limit of $g$. I know that b) has something to do with Cauchy's formula, and I tried to use the RHS to get the LHS, but got stuck after expanding it. I also tried ...
2
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3answers
50 views

Does this equation have a complex number solution?

Does this equation have any solutions? I know it does not have any real number solutions, but how about complex number solutions? $$\sqrt{z^2+z-7}=\sqrt{z-3}$$ I understand that when you solve this ...
2
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0answers
15 views

Quadratic Taylor coefficient for free by normalizing; when is second Taylor coefficient real?

Let $f: \mathbb{R} \to \mathbb{C}$ and suppose that we know $|f(\lambda)| = 1$ for all $\lambda$. Consider the Taylor series around $0$: $$ f(\lambda) = a + b\lambda + c\lambda^2 + \cdots. $$ Instead ...
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2answers
31 views

Determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=27i$

I've just learned complex numbers in Mathematical Analysis 1, and I'm stuck in the following problem: I would like to determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=...
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0answers
24 views

Prove that $||z|-|w||≤|z-w|$

I need to prove $||z|-|w||≤|z-w|$, possibly using $|z+w|≤|z|+|w|$ I have labeled z as $z=a+bi$ and w as $w=c+di$. Got to the point where I have $|\sqrt{a^2+b^2}-\sqrt{c^2+d^2}|≤\sqrt{(a-c)^2+(b-d)^2}$...
2
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3answers
76 views

Does $\sqrt{i^4} = i^2$?

I'm assuming it doesn't, because if it did, then $1 = \sqrt{1} = \sqrt{i^4} = i^2 = -1$. In general, does $\sqrt{x^4} = x^2$?
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1answer
32 views

Summation from a real number to a complex number

What would the solution of this be: $\displaystyle \sum_{n=1}^i n$ where $i$ is the imaginary unit. or any other formula really, I'm just interested to know how would a summation work from a real ...
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1answer
35 views

Finding an argument of sum of two complex hyperbolic functions

I want to find the absolute value and the argument of the following complex number $$ -\cosh[\sqrt{z_1}]+\frac{z_2}{\sqrt{z_1}}\sinh[\sqrt{z_1}] $$ where $z_1$ and $z_2$ are two complex numbers. Can ...
0
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1answer
29 views

Can any complex equation be seperated into two equations? [on hold]

Considering that √-1 is i, but negative numbers are actually positive numbers that indicate an opposite direction or quantity, can any complex equation be seperated into two positive equations? If so, ...
2
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1answer
32 views

Poles of $f(z)=\frac{\sin(\frac{\pi z}{2})}{\sin(\pi z)}$

Consider the complex valued function $$f(z)=\frac{\sin(\frac{\pi z}{2})}{\sin(\pi z)}$$ I am trying to investigate the poles at integers points. That is points where $z=n,n\in \mathbb{Z}$ Case 1: ...
0
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1answer
24 views

Complex logarithm multiplication/exponent rule

I am trying to show that $\log(i^2) \ne 2\log(i)$ and that $\log(i^{1/2}) = 1/2\log(i)$ but I am having difficulties. I know that $\log(z) = \ln|z| + i\arg(z)$ but when playing around with these ...
6
votes
2answers
46 views

Prove that $\frac{(z_1 + z_2)(z_2 + z_3)…(z_{n-1} + z_n)(z_n + z_1)}{z_1 \cdot z_2 \cdot … \cdot z_n}$ is real

$z_1, z_2, ... z_n$ are complex numbers such that $|z_1| = |z_2| = ... = |z_n|$. How to prove that $\frac{(z_1 + z_2)(z_2 + z_3)...(z_{n-1} + z_n)(z_n + z_1)}{z_1 \cdot z_2 \cdot ... \cdot z_n}$ is ...
0
votes
1answer
33 views

Is there an infinite number of number sets? (natural, integers, real, complex…) [on hold]

I've heard of a number set on that's the smallest set after complex numbers, according to http://www.askamathematician.com/2012/09/q-is-there-a-number-set-that-is-above-complex-numbers/ (The Physicist)...
4
votes
3answers
107 views
+50

Show that $e^z$ is continuous on $\mathbb{C}$

I know that $e^z$ is continuous on $\mathbb{R}$, but how would I show this rigorously on $\mathbb{C}$ using the $\epsilon - \delta$ definition of continuity? I know how to begin: If $|z - z_0| < \...
4
votes
2answers
59 views

minimum value of $\bigg||z_{1}|-|z_{2}|\bigg|$

If $z_{1}\;,z_{2}$ are two complex number $(|z_{1}|\neq |z_{2}|)$ satisfying $\bigg||z_{1}|-4\bigg|+\bigg||z_{2}|-4\bigg|=|z_{1}|+|z_{2}|$ $=\bigg||z_{1}|-3\bigg|+\bigg||z_{2}|-3\bigg|.$Then ...
1
vote
1answer
37 views

How to solve a two variable equation

Suppose $m,n$ are positive constants,there is an equation $m|\lambda_1|^2+|\lambda_2|^2-n\lambda_1\bar{\lambda_2}-n\lambda_2\bar{\lambda_1}=0$,where $\lambda_1,\lambda_2$ are nonzero complex numbers. ...