Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
11,619 questions
1
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0answers
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 ...
1
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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-
$$\tan \phi_{m+n} = \tan \phi_m \sec \...
4
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0answers
36 views
$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})…(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ . Find $P$.
$$(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)$$ where $\omega_n$ is the nth root of unity.
Find ...
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0answers
13 views
Using rules of spiral enlargement.
If I may summarize the steps used in this simplification:
They express the spiral enlargement in terms of a polar complex number.
They multiply this result with $\vec{AT}$ to get $\vec{AW}$.
To find ...
0
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0answers
14 views
What is the range of koebe function? [on hold]
I don't know about the behaviour of koebe function k(z)=z/(1-z)^2.
1
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1answer
49 views
Show that $\oint_{\gamma} \frac{Log z}{z}dz=0$
I need to prove that
$$\oint_{\gamma}\frac{\log z}{z}dz=0$$
where $\log z= \log |z|+ \text{Arg } z$ ($\text {Arg}$ is the principal argument), $\gamma:z=e^{it},\quad 0\leq t\leq 2\pi$.
I know ...
1
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0answers
23 views
Analytics function satisfied certain equality
Question: If $f(z)$ is analytic in a domain $D$, $f'(z)\neq 0$, $|f(z)|\neq 1$, then show that $$w=log\frac{|f'(z)|}{1-|f(z)|^2}$$
satisfy $\nabla^2w=4e^{2w}$.
I don't know how to start with this ...
2
votes
4answers
174 views
Why is $e^{2πi} = e^0$ true while $2πi = 0$ is false?
I came across a perplexing thing while testing some assumptions in Wolfram|Alpha, and as I don't have a math education beyond college algebra, I thought this would be a good place to ask. I would just ...
-2
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3answers
90 views
Is $i = e^{\frac{\pi}{2}e^{\frac{\pi}{2}^{.^{.^.}}}}$? [on hold]
I came up with this and I am wondering if it is true, because it seems illogical that $i$ can be made from an infinite power tower of reals.
The way I found this is the following:
$$i=e^{\frac{\pi}{2}...
1
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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
votes
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
votes
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}{...
0
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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
10 views
For each part, give three different complex numbers that satisfy the equation. Enter them in the form a+b*I (with a capital I) separated by commas. [on hold]
For each part, give three different complex numbers that satisfy the equation.
|z-5|=|z-4i|
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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 ...
0
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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 ...
0
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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
votes
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 ...
0
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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
votes
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.
0
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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 $z=a+bi~$:
$$\theta=\tan^{-1}(b/a),~~a>0;~~~ \theta=\pi+\tan^{-1}(b/a),~~a<0$$
...
2
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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:
$$\int^0_{-\pi} {ie^{2it}\sin(2t)}dt = ...
1
vote
1answer
24 views
Using the fifth root of unity to show the cosine equation
Consider the equation expressed the fifth root of unity: $z^5-1=0$
To show that: $$2(\cos(\frac{2\pi}{5})+\cos(\frac{4\pi}{5}))=-1\\4\cos(\frac{2\pi}{5})\cos(\frac{4\pi}{5})=-1$$
I have already ...
3
votes
0answers
24 views
Generalizations of the field concept to triplets of elements
In the 'Field' article at MathWorld there is a phrase that I don't understand:
It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements ...
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0answers
27 views
0
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1answer
53 views
Doesn't $i=-i$ from the definition $i^2=-1$? [duplicate]
There is 2 definition about imaginary numbers:-
Imaginary unit, $i=\sqrt{-1}$
The square of Imaginary unit, $i^2=-1$
But the later is used mostly, as known to me, because of many reasons. And my ...
2
votes
1answer
48 views
Let $f$ be the disk $D[0, 1]$ be a holomorphic function. Show the following
I was able to prove a) using the limit of $g$. I know that b) has something to do with Cauchy's formula, and I tried to use the RHS to get the LHS, but got stuck after expanding it.
I also tried ...
2
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3answers
50 views
Does this equation have a complex number solution?
Does this equation have any solutions? I know it does not have any real number solutions, but how about complex number solutions?
$$\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. $$
Instead ...
3
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2answers
31 views
Determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=27i$
I've just learned complex numbers in Mathematical Analysis 1, and I'm stuck in the following problem: I would like to determine the number of point $z \in \mathbb{C}$ such that $(2z+i\overline{z})^3=...
0
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0answers
24 views
Prove that $||z|-|w||≤|z-w|$
I need to prove $||z|-|w||≤|z-w|$, possibly using $|z+w|≤|z|+|w|$
I have labeled z as $z=a+bi$ and w as $w=c+di$.
Got to the point where I have $|\sqrt{a^2+b^2}-\sqrt{c^2+d^2}|≤\sqrt{(a-c)^2+(b-d)^2}$...
2
votes
3answers
76 views
Does $\sqrt{i^4} = i^2$?
I'm assuming it doesn't, because if it did, then $1 = \sqrt{1} = \sqrt{i^4} = i^2 = -1$. In general, does $\sqrt{x^4} = x^2$?
0
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1answer
32 views
Summation from a real number to a complex number
What would the solution of this be:
$\displaystyle \sum_{n=1}^i n$
where $i$ is the imaginary unit. or any other formula really, I'm just interested to know how would a summation work from a real ...
-1
votes
1answer
35 views
Finding an argument of sum of two complex hyperbolic functions
I want to find the absolute value and the argument of the following complex number
$$
-\cosh[\sqrt{z_1}]+\frac{z_2}{\sqrt{z_1}}\sinh[\sqrt{z_1}]
$$
where $z_1$ and $z_2$ are two complex numbers. Can ...
0
votes
1answer
29 views
Can any complex equation be seperated into two equations? [on hold]
Considering that √-1 is i, but negative numbers are actually positive numbers that indicate an opposite direction or quantity, can any complex equation be seperated into two positive equations? If so, ...
2
votes
1answer
32 views
Poles of $f(z)=\frac{\sin(\frac{\pi z}{2})}{\sin(\pi z)}$
Consider the complex valued function
$$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
votes
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.
...
