Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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1answer
24 views

Geometric meaning of $z=\frac{x_1+x_2}{2}$, $y=\frac{x_1+x_2}{|x_1+x_2|}$, and of a substitution (in the complex plane)

Let $S$ be a unitary (that is, $r=1$) circle centered in the origin of the complex plane. Let $x_1 \in S$ and $x_2 \in S$ be the vertices of the triangle $Ox_1x_2$. What is the geometric meaning of ...
5
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3answers
145 views

If $z_{n+1}=\frac{27}{\overline{z_{n}}}+6$ and $z_1 = 3 + 6i$, then find $z_{n}$

Let the complex sequence $\{z_{n}\}$ satisfy $z_{1}=3+6i$, and $$z_{n+1}=\dfrac{27}{\overline{z_{n}}}+6.$$ Find the $z_{n}$. My idea: since ...
0
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1answer
11 views

Draw a set of values in complex plane

I need help with that exuations: a) $\arg (z+2-i)=\pi$ b) $\pi \leqslant \arg [(-1+i)z]\leqslant \frac{3 \pi}{2}$ c) $ \arg \frac {i}{z}=\frac{3\pi}{4}$ I have more of them but I don't know how to ...
0
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1answer
21 views

Linearly Independent or Dependence of Complex Vectors — Homework Help [on hold]

Determine whether the indicated sets of complex vectors are linearly independent or dependent. $\left[\begin{array}{cc}i\\1\end{array}\right]$$\left[\begin{array}{cc}1\\i\end{array}\right]$ ...
1
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0answers
2 views

Weighted undirected graphs, complex Laplacian, complex eigenvalues & spectral clusering

I am rather puzzled and confused, I have been trying to get a clear understanding of how would spectral clustering work for an undirected weighted graph, I have used the normalized Laplacian, but I ...
1
vote
3answers
31 views

Calculate value of a real number, considering “n” as a natural number

How could I calculate the value of the real number: $$ (1 +i \sqrt{3})^n + (1 - i \sqrt{3})^n $$ ...considering $n$ as a natural number and $i$ as the imaginary unit.
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1answer
22 views

$\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$\{0}

I need to prove the set identity of the complex logarithm $\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$. ...
0
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1answer
27 views

Calculate complex number considering…

How can I calculate: $$ \frac{1-Z}{1+Z} $$ ...considering $Z = \cos(\alpha) + i \sin (\alpha)$ I have replaced the expression but I don't know how can I continue...
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1answer
13 views

the crossover point of four complex points

If there is four complex points $z_1,z_2,z_3,z_4$ in complex plane $\mathbb{C}$, I want to get the crossover point of the line $z_1z_2$ and $z_3z_4$. If I use the $Re(z_i)$ and $Im(z_i)$, it is easy ...
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2answers
63 views

What is meant by $(a+ib)^{c+id}$

I am currently studying complex numbers. Recently I saw terms like this: $(a+ib)^{c+id}$ , actually, I was simplifying them. But it was okay till I arrived at the following: ...
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1answer
19 views

Connectedness of a set of complex numbers

Is the graph of {$xy=1$} in $\mathbb C^2$ connected? I know $xy=1$ is disconnected in $\mathbb R^2$ by drawing its graph.But how to approach in $C^2$
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5answers
51 views

How to find the complex solution of $x^6$

How do you find the complex solutions to $x^6+x^3-2=0 $ Obviously $x=1$ is one solution, but i cant get further than that.
1
vote
1answer
13 views

Complex number modulus identity (on unit circle)

For any three complex numbers $z_1, z_2, z_3$ on the unit circle, $|z_1 + z_2 + z_3| = |z_1 z_2 + z_1 z_3 + z_2 z_3|$. I am able to prove this by putting each number in modulus-argument form and ...
0
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0answers
17 views

Existence of a root $\alpha$ so that $|\alpha+i| <1$

For some monic polynomial $P(x) = \displaystyle \sum_{k=0}^n a_k z^k, 0 < |P(i)| < 1, a_k \in \mathbb{R}, k=0,1,...,n$, how does one show that a complex root $\alpha$ exists such that $|\alpha + ...
0
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1answer
20 views

Complex Cubic Equation z^3+3z+2i=0

How we can solve the equation $z^3+3z+2i=0$ ? And is there exist a general method to solve similar equation?
2
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0answers
47 views

For which $m$ is this sum of roots of unity $0$?

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
0
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1answer
23 views

Prove that $\hat{f}(n)={\frac{2}{\pi(1-4n^2)}}$, given that $f(x)=|sin(\pi x)|$

Prove that $$\hat{f}(n)={\frac{2}{\pi(1-4n^2)}},\ given\ thatf(x)=|sin(\pi x)||,\int_{0}^{1}sin\pi(x)dx={\frac{2}\pi}\\where\ \hat{f}(x)=\int_{0}^{1}f(x)e(-nx)dx. \ Use\ the\ fact\ ...
0
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1answer
45 views

If $w = e^{2i\pi/5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$=?

If $w = e^{i\frac{2\pi}5} $, then $1 + w + w^{2} + w^{3} + 5w^{4} + 4w^{5} + 4w^{6} + 4w^{7} + 4w^{8} + 5w^{9}$ =? I substituted $w$ into the expression and combined similar terms. I then tried to ...
5
votes
6answers
1k views

If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$

If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$ Hint: solve for $b^2$ in terms of $a^2$ and then solve for $a$ I've attempted the question but I don't think I've done it correctly: $$ ...
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2answers
31 views

Write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form $a+bi$

I have to write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form: $a+bi$. I think I should consider $z=\cos(k \pi / 6)+i\sin(k \pi / 6)$ and also use the fact that $\sum ...
1
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3answers
21 views

summation and product of sin and cos

I wonder how to find summation for $\displaystyle \sum_{k=0}^{n-1}(\cos{\frac{2\pi k}{n}+i \sin\frac{2\pi k}{n}})$ and the same for product $\displaystyle \prod_{k=0}^{n-1}(cos{\frac{2\pi k}{n}+i ...
0
votes
1answer
15 views

A question on absolute values of line integrals

Let $f:U\rightarrow \mathbb{C}$ be continuous and $\gamma:[a,b]\rightarrow U$ be a smooth path where $U$ is open. Then we know that $$\int_{\gamma}^{}\ f(z)dz=\int_{\gamma}^{}\ ...
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3answers
65 views

$\int_{|z+1|=2}^{}\frac{e^z}{z-2}dz \leq2\pi e$

This is how I attempted to solve this but I could not get the exact inequality. $\gamma(t)=1+2e^{it}, t\in [0,2\pi],f(z)=\frac{e^z}{z-2}$ $|e^{1+2it}|=e|e^{2it}|=e.|e^{it}|.|e^{it}|=e$ ...
0
votes
1answer
15 views

A question on branch of an inverse

Ignoring the 1st part of the 1st sentence of the question all I want to get is a branch $f$ of the inverse function of $g(z)=z^4. $ This is how I set about doing it, however, I need to verify this. ...
1
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1answer
87 views

Is the series: $\,\sum_{n=1}^{\infty}\frac{\mathrm{e}^{-in}}{n}\,$ divergent?

According to mathematica, the complex series $\displaystyle\sum_{n=1}^{\infty}\frac{e^{-in}}{n}$ does not converge. I know that the factor $\dfrac{1}{n}$ in the above series is diverging, but I don't ...
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2answers
32 views

Limit of $i^{n!}-2^{-n}$

I ran into this problem in Palka's Book which said to compute the limit of $z_n=i^{n!}+2^{-n}$. My approach was to consider the real and imaginary limits separately. Clearly the limit of the real part ...
0
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1answer
14 views

Lower bound of integral (difficult one)

Find an upper bound for $|\int_C(\cos z+\frac{e^z}{z+1})dz|$, where C is the circle $|z+i|=4$ Find a lower bound for $|\int_C(\cos z+\frac{e^z}{z+1})dz|$, where C is the circle $|z|=4$ The integral ...
1
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4answers
138 views

Why is $i^2$ equal to $-1$? [duplicate]

In this KhanAcademy link at 2:25, Sal (the narrator) says that $i^2$ is negative 1 and he didn't explain why. Why is this so? What is the intuition behind it?
2
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1answer
53 views

Special feature of the function f(z) = $|i + z|^2 + az + 3$

I have to solve following problem: Find all the values of a (a is a real number) that the function f : $f(z) = |i + z|^2 + az + 3$ (z is a complex number, i is an imaginary unit) has a following ...
3
votes
0answers
32 views

Is the exponential function the one this problem is hinting at?

Suppose that $f$ is holomorphic on all of $\mathbb{C}$ and that $$\lim_{n\rightarrow \infty} \left(\frac{\partial}{\partial z}\right)^nf(z)$$ exists, uniformly on compact sets, and that this limit ...
1
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2answers
32 views

Determining the branch of logarithm

I want to determine a branch of logarithm such that $f(z)=L(z^3-2)$ is analytic at $0$. I am not really sure how to find a branch but I will explain few things I tried. Since $z^3-2$ maps $0$ onto ...
0
votes
3answers
30 views

Verify Euler's formula

Verify Euler's formula for $e^{ix}$ by considering $\frac{dz}{dx}$ where $z=r(\cos x+i\sin x)$ I tried taking the derivative of z but could not get to Euler's from there.
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0answers
24 views

Consider$\ f$, defined for all complex numbers except$\ x_0$. What can(not) happen to the real part of$\ f\left(x_0+bi\right)$ as$\ b\to0$?

Yes,$\ f$ is holomorphic. In other words, given that$$\ \lim_{x\to x_0}f(x)=\infty,$$what do we know a priori about $\ \lim_{b\to0}\Re\left(f\left(x_0+bi\right)\right)$? Can it be any extended real ...
1
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2answers
33 views

Finding the fourth roots of $\,5(\cos(3)+i\sin(3)).$

Find the four fourth roots of $\,5(\cos(3)+i\sin(3)).$ I tried to convert to polar form so I could set up an equation like $\,x^4=5e^{i3},\,$ but I am unsure to continue.
0
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1answer
18 views

Half Angle Trig and Complex Numbers

By making use of the half-angle formulae, or otherwise, prove that $$\frac{1+\cos x+i\sin x}{1-\cos x+i\sin x}=\cot{\frac x2} e^{i(x-\frac\pi2)}$$
1
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2answers
37 views

Proof for $|1 - z| \geq 1 - |z|$ for $|z| < 1$, $z \in \mathbb{C}$

I can prove it "by picture" by drawing a picture of a circle of radius $|z|$ centered at $(0, 1)$. Then $1 - |z|$ is the length from the origin to the intersection of the circle with the x-axis (to ...
0
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1answer
34 views

evaluating Pi with imaginairy unit i leads to contradiction!

I was reading about evaluating $i^i$ so I tried that with Mathematica and got a real (R) result and Mathematica suggested an alternative form being $e^{-pi/2}$ so I solved for $\pi$: $i^i = ...
3
votes
2answers
86 views

A double Summation involving 7th roots of unity

Is there possibly a closed form for $$\sum_{m=1}^{\infty} \left(\sum_{k=1}^{6} \dfrac{1}{m-\alpha^k}\right)^2$$ where $\alpha=e^{2\pi i/7}$ ? I'm having problems evaluating the first sum, let alone ...
0
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1answer
11 views

Convert a complex number into trigonometric form.

Convert the complex number $$\Large e^{\frac{iz}{\bar z+1}}$$ $z\in \mathbb C$, into trigonometric form. any suggestions please?
1
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1answer
23 views

Evaluate and find the principal value of $(-1+i)^ {2-i}$

Can anyone please help me evaluate and find the principal value of $(-1+i)^{2-i}$ I got up to $=e^{2-i}(ln(-1+i))$ $=e^{(2-i)(1/2 ln(2)+i(3pi/4))}$
0
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0answers
22 views

Complex numbers function

I need advice solving a problem. Let $f$ be function of variable $z$, which transforms the complex plain in a specific way: Homothetic transformation of factor $2$ Translation $1$ right and $2$ ...
0
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0answers
32 views

Properties of a real function defined implicitly by an equation with $\int_0^x e^{f(t)}\mathrm{d}t+i x\int_0^1 e^{f(x-xt)}\mathrm{d}t$

Let $f : \mathbb{R_{\geq 0}} \rightarrow \mathbb{R_{\geq 0}}$ and $\forall x$ let a complex $z$ such that; $$z=\int_0^x e^{f(t)}\mathrm{d}t+i x\int_0^1 e^{f(x-xt)}\mathrm{d}t$$ and ...
-3
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2answers
49 views

How to evaluate this complex exponential number? [closed]

Does anyone know how to conjugate $z=e^{0.5 \ln(-1-i)}$ Is it i need to compare with $z=re^i$?
6
votes
1answer
125 views

Are there any arguments against the Riemann hypothesis?

We all know the well known Riemann hypothesis that the zeroes of the Riemann-zeta function have real part $1/2$ seems to hold (as far as I know) for all prime numbers. I was curious if there were any ...
1
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2answers
48 views

If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z $ is real, what is $z$?

Two questions from Grade Twelve class on complex numbers If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z$ is real, what is $z$? How do you solve graphically given the ...
0
votes
0answers
22 views

Find the derivative of $exp[(1+2z+z^2)^{-1}]$

Find the derivative of $exp[(1+2z+z^2)^{-1}]$ Well, I wanted to do it this way: $exp[(1+2z+z^2)^{-1}] = e^{1 \over (1+2z+z^2)}$ Now we have $exp'[(1+2z+z^2)^{-1}] = (e^{1 \over (1+2z+z^2)})'$ ...
1
vote
1answer
25 views

Branches of the square root function in the domain $D=\mathbb{C}$\ $[0,\infty)$

I saw the solution for this in Palka's book and one of the branches was defined as follows. $$ g(z) = \begin{cases} \sqrt{z}, & z\in D ,Im(z)\geq0 \\ -\sqrt{z}, & z\in D ,Im(z)<0 ...
0
votes
1answer
31 views

Show that $\frac{1}{2\pi} \int_0^{2\pi} e^{ik\theta}d\theta = 1$ or $0$ depending on $k$.

I'm asked the problem (restating from the question title), $$\textrm{Show that }\frac{1}{2\pi} \int_0^{2\pi} e^{ik\theta}d\theta = \begin{cases} 1,\, k=0\\ 0, \, k\neq0 \end{cases}$$ My Attempt: ...
-1
votes
0answers
38 views

cauchy-riemann equation in complex analysis [closed]

Hi can someone help me with this problem. Verify the function below satisfies the Cauchy-Riemann equations and determine the derivative, $$f(z) = {e^{{z^2}}}.$$
1
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0answers
15 views

Branch of the nth root of a complex number

Let $n\geq2$ and $f(z)=z^n,z\in \mathbb{C}$. I need to show that if $L$ is a branch of the logarithm function in a domain $D$ then $h_{1/n}(z)=e^{L(z)\over n}$ is a branch of $f^{-1}$ in $D$. (If $L$ ...