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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1
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2answers
70 views

Finding the minimum value of magnitude of this complex number

|z|>2 then find the minimum value of "|z+1/2|" How can I solve this using circles?
-1
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2answers
70 views

If $\sum (2009-r)\cos(\frac{2\pi r}{2009})=-n/2;\quad 1\leq r\leq 2008,$ then the digits in the unit's …

$$\text{If}\;\sum (2009-r)\cos\left(\frac{2\pi r}{2009}\right)=-n/2;\quad 1\leq r\leq 2008$$ then the digits in the unit's place of $(9417709487)^n$ must be equal to? Well how do I proceed? Hints ...
3
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2answers
69 views

Let $A_1,A_2,..,A_n$ be the vertices of n sides of a regular polygon such that $1/A_1.1/A_2=1/A_1.1/A_3+1/A_1.1/A_4$ then value of $n$ must be?

Let $A_1,A_2,..,A_n$ be the vertices of n sides of a regular polygon such that $$\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$$ then value of $n$ must be? Any ideas on how to start? I'm having ...
0
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3answers
52 views

Trigonometry-Complex Numbers Based Problem

If $2^7\cos^5x * \sin^3x$=$a\sin8x- b\sin 6x +c\sin 4x + d\sin 2x$ where $x$ is real then what will be the value of $a^4 + b^4 + c^4 + d^4$? Even a hint will suffice... I don't know how to proceed! I ...
0
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0answers
97 views

Does there exist an entire function such that $f\left(n+\frac{1}{n}\right)=0$

Does there exist an entire function $f:\mathbb C \to \mathbb C$ such that $f\left(n+\frac{1}{n}\right)=0$ for all $n\in \mathbb N$ ? I tried through Taylor series expansion , also by contradictory ...
1
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2answers
63 views

Complex Number -A problem on conjugate

|$z_1$|=2,|$z_2$|=3,|$z_3$|=4 and |$2z_1+3z_2+4z_3$|=9 then the absolute value of $8z_2z_3+27z_3z_1+64z_1z_2$ must be equal to? ($z_1,z_2,z_3$ are complex numbers) I tried manipulating with the ...
3
votes
2answers
76 views

Complex number, how to solve

Calculate i)$(1+i)^i$ ii)$(-1)^{\frac{1}{\pi}}$ I did i)$(1+i)=\sqrt{2}e^{i\frac{\pi}{4}}$. Knowing that if $z$ and $c$ are complex numbers $z^c=e^{c\log z}$ $$(1+i)^i=e^{i\log(1+i)}=i\log(1+i)=i\...
4
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2answers
445 views

Base conversion: How to convert between Decimal and a Complex base?

My motivation for this question is exploring beyond the ideas in Project Euler Problem 508. In that problem, it is helpful to know how to convert between a decimal number and a number in base $(-1+i)$....
1
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1answer
30 views

Complex number, logarithm power proof

Proof that i)$Log(1+i)^2=2*Log(1+i)$ ii)$Log(-1+i)^2\neq2*Log(-1+i)$ What I did i)By definition $z^a=e^{a\log z}$, so if $z=(1+i)$ and $a=2$ $$Log(1+i)^2=Log(e^{2\log(i+1)})=2*log(i+1)$$ But I do ...
1
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1answer
19 views

Find all solutions for a complex logarithm

$\log z = 6i$ I am working on a problem very similar. What I am seeing $\log z = \ln|z| + i(\theta + 2\pi n)$ for $n\in\mathbb{Z}$ What I am curious about, as if seen obvious to me that $ \log ...
0
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1answer
28 views

Complex number, logarithm and exponential

Find i)$Log(-ei)$ ii)$Log(1-i)$ I'm not too sure about how to solve this, what I did is Take $z=-ei$ so $Log(z)=\log r + i\theta, \space r>0, \space -\pi<\theta\leq\pi$ $$r=\sqrt{0+e^2}=e\...
0
votes
1answer
41 views

Complex number, logarithm

Find i)log(e) ii)log(i) I do not know if these issues are of simple fact, that there is something behind. I did i)Since $log$ and $e$ are inverse functions so$$log(e)=log(e^1)=1$$ Knowing that $z$...
1
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0answers
25 views

Does there exists an automorphism of $\Bbb{C}$ that's also an exponential hom?

Is there an automorphism of the field $\Bbb{C}$ of complex numbers, $\phi$, such that for all $z, w \in \Bbb{C}$ we have in addition to being a ring hom, $\phi(z^w) = \phi(z)^{\phi(w)}$?
0
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0answers
30 views

Complex number, exponential

Find all values of $z$ such that i) $e^z=-2$ ii) $e^z=1+i\sqrt{3}$ iii) $e^{2z-1}=1$ What I did I know that $e^z=e^x(cosy+iseny)$, then i) $e^x\geq0 \forall x \in \mathbb{R}$ so I need to find $...
0
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1answer
52 views

Line integral confirmation and Geometric interpretation

I have $$\int_{C}(z - \bar{z})dz$$ where $C = \{z \; : \; |z-1| =2\}$ So I parametrize $C$ by letting $z = 2e^{it} + 1 = 2\cos(t)+ 1 + 2i\sin(t)$ and let $x = 2\cos(t)+1$ and $y = 2\sin(t)$, for $t \...
3
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3answers
114 views

Corrrect treatment of a limit approaching zero in complex plane?

How to (correctly) evaluate this limit? $$\lim_{k\to0}\left(\frac{\mathrm{i}+k}{\mathrm{i}-k}\right)^{-\frac{\mathrm{i}}{k}}$$ Here $\mathrm{i}$ is the imaginary unit. Mathematica gives $1/e^2$ (...
1
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2answers
47 views

Show that $\frac{z}{z+1} = 1 - \frac{1}{z+1}$

I'm doing some complex analysis work and I came upon this equivalency: $$f(z) = \frac{z}{z+1} = 1 - \frac{1}{z+1}$$ I was trying to find out how to go from $z/ (z+1)$ to the equivalent expression ...
0
votes
1answer
22 views

Prove that $|f^{(k)}(0)|/k! \leq M (e/k)^k$

Assume that f(z) is an entire function and $|f(z)| \leq Me^{|z|}$. Here's what I have gotten so far. I used the Cauchy estimate which is really simple math and I've gotten it down to something like $...
1
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2answers
251 views

Find the greatest value of $\arg z$ achieved on a circle in $\mathbb{C}$

Consider the circle $$|z-6i|=3;$$ its centre is $(0,6)$ and its radius is $3$. I want to find the greatest value of $\arg z$ achieved on this circle. My idea is that the tangents to the circle from ...
0
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2answers
23 views

Prove that for all $z\in\mathbb{C}$, if $|z| = 1$ and $z\neq−1$ then $Re((1-z)/(1+z)) = 0$

What I have so far: Assume $|z|= 1$ and $z\neq-1$, then $z=1$ or $z=i$ or $z=-i$. If $z=1$, $Re(1-1)= 0$ as needed, but for trying to prove $z=i$ and $z=-i$ I get $Re(1-i) = 1$ and vice versa... ...
4
votes
2answers
41 views

Euclidean norm of complex vectors

I am working on a proof: One has two vectors, $u,v \in \mathbb C^n$, such that $u \cdot v=0$ . I am trying to prove that $$|u + v|^2 = |u|^2 + |v|^2.$$ I am a little stuck on how to do $u + v$ ...
1
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1answer
16 views

Proving that $|a(z)^\alpha - b(z)^{\alpha}| \rightarrow 0.$

Related to an earlier question, I asked that Is it true that $|a^{\alpha} - b^{\alpha}| \leq |a-b|^{\alpha}$? I was asking this since I am given that two functions $a(z)$ an $b(z)$ satisfy $|a(z)- b(...
5
votes
1answer
134 views

Writing circles as $|z-a| = \lambda |z-b|$ for the same $a,b$

My problem is in the context of the complex plane. I want to know if given two disjoint, not concentric circles $C_1,C_2\subset \mathbb{C}$, can you find $a,b\in \mathbb{C}$ such that $$C_1=\{z\in \...
1
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1answer
45 views

Complex integration confusion

I wish to compute $\int_{C}(x^2 - iy^2)dz$, where $C := \{z\mid |z|=1\}$ is positively oriented. I am a bit confused on what $dz$ actually is. I know I have $\int_{C}x^2dz - i\int_C y^2dz$, but I ...
1
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1answer
61 views

Solve $(z+2) /( z-3i) = 4+2i$ for $z$ in complex numbers

I'm having some trouble trying to isolate $z$. Can I multiply $(4+2i)$ by $(z-3i)$ without changing $z$ into standard form?
0
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3answers
57 views

prove that this complex function is the zero function

we know that $f$ is analytic in the ring $R=$ {$z: r_1 < |z| < r_2$}. moreover at the circle {$z:|z|=r_2$} $f$ is continuous and for every $z$ in that circle $f(z) = 0$ . We have to prove that $...
0
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2answers
29 views

is the following complex function is defined at deleted neighborhood of $z=0$

the function is: $\dfrac{2\pi z - i}{\sinh(\frac{1}{z})}$ of course the function is not defined at $z=0$, but what happen in a deleted neighborhood of that point ?
2
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2answers
45 views

Prove $\frac{1}{z^2}=\sum\limits_{n\ge0}(-1)^n(n+1)(z-1)^n$

Prove that for any complex number $z$ such that $|z-1|<1$, one has: $\frac{1}{z^2}=\sum\limits_{n\ge0}(-1)^n(n+1)(z-1)^n$ What I've done; $\displaystyle\frac{1}{z^2}=\left(\frac{1}{z}\right)...
2
votes
3answers
114 views

Does $\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0 $ for all $n$ imply that $\lambda_1= \lambda_2= \dots= \lambda_k = 0 $?

Suppose $\lambda_1, \lambda_2, \dots, \lambda_k $ are complex numbers that $\forall n \in \mathbb{N}$ satisfy $$\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0.$$ Can we deduce that $\lambda_1= \...
0
votes
2answers
67 views

Positively oriented circle

I wish to have a positively circle centered around $z_0 = -1 +2i$ with radius $3$. Clearly, the circle is $|z - z_0| = \rho = 3$. And written out we have $ \sqrt{(x + 1)^2 + (y - 2)^2} = 3$. I am ...
0
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3answers
37 views

Basic Expansion

I know this is probably easy but how would I multiply out $$(z-(5+i))(z-(5-i))$$ Would I multiply inside both brackets by the "-" then the z part? Thanks
0
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2answers
33 views

different ways of calculating laurent series of $\frac{1}{1+z^2}$ at $z=0$

it seem to me that in that simple case there are different ways of doing this. I have to wonder on the general case of $\dfrac{1}{a+z^2}$ at $z=0$. edit: the interesting part of course is where $|z|...
0
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2answers
39 views

Verify that Log$(z^{w}) = w$Log$z$ + $2\pi i n$

The symbol "Log" denotes the complex logarithm. Let $w$ be a complex number so that $w = u+iv$ for some reals $u, v.$ We have $$\mbox{Log}(z^{w}) = \log |z^{w}| + i\arg (z^{w}) = u\log |z| - v\arg (...
0
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2answers
63 views

Principle argument for $-1-i=-\frac{3\pi}{4} \ne \frac{\pi}{4}$

I want to find the principle argument for $-1-i$ Using the standard method for finding $\theta=\arctan\left({\frac{y}{x}}\right)$ $$\arctan(\frac{-1}{-1})=\frac{\pi}{4}$$ But: $$\operatorname{Arg}(...
1
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2answers
32 views

Proving a property of cubic root of unity.

Can anyone help on proving below expression? For $w$(complex cubic root of unity), $1+w^r+w^2 r = 0$ for any positive integer $r$, which is not a multiple of $3$.
2
votes
2answers
86 views

What is the correct value?

My confusion is: $(-9)^{2/3} = ((-9)^{2})^{1/3} = ((-9)^{(1/3)})^{2} = 4.32$ But my calculator shows math error, and google says: $(-9)^{2/3} = 2.16+3.74i$
2
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1answer
19 views

Show that for any $r>0$ and $z \in \mathbb{C}$ we have that $B_r(z)\subset S_r(z)$

Show that for any $r>0$ and $z \in \mathbb{C}$ we have that $B_r(z)\subset S_r(z)$ In other words, that any Ball of radius $r>0$ centred at $z \in \mathbb{C}$ is a subset of the square $S_r(...
4
votes
3answers
621 views

Modulus of a complex number

I am now encountering a problem regarding on complex analysis Lets say we have $w=u+iv$ What would it be for $$|w|^{2}$$ I check a lot of videos and lecture notes, and realize the answer is $$u^{2}+...
1
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1answer
48 views

help on proving some property of cubic root of unity

Anyone can help to prove below expression $\omega^r$ and $\omega^{2r}$ satisfy the equation $x^3 - 1=0$ for any positive integer $r$, where $\omega$ and $\omega^2$ are non-real cube roots of ...
2
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2answers
52 views

If a, b are complex numbers then the maximum value of $\dfrac{a\bar b+\bar ab}{|ab|}$

If a, b are complex numbers then the maximum value of $\dfrac{a\bar b+\bar ab}{|ab|}$ is (A) 2 (B) 1 (C) the expression may not always be a real number and hence maximum does not make sense (D) ...
1
vote
2answers
40 views

Show that $e^\mathbf{iA} + e^\mathbf{iB} = 2e^\frac{i(A+B)}{2}\cos(\frac{A-B}{2})$

Where $i=\sqrt{-1}$ and $A,B\in \mathbb{R}$ are constants. I've tried already with Euler's formula, but cannot prove the equation above. Best Regards, Thanks.
1
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1answer
48 views

Problem in a proof that we cannot order complex numbers

The order axioms of real numbers state 1) Either $x = y$ or $x < y$ or $x > y;$ 2) If $x < y,$ then $x+z < y+z;$ 3) If $x, y > 0,$ then $xy > 0;$ 4) If $x > y$ and $y > z,$ ...
0
votes
0answers
36 views

Indeterminacy of complex numbers

If $f(z)=\frac{|z|(1+z^2)^4}{(z+1)^2}$ calculate $f'(0)$ I did: $g(z)=|z|\rightarrow g'(z)=\frac{z}{|z|}$ and $h(z)=\frac{(1+z^2)^4}{(z+1)^2}\rightarrow h'(z)=\frac{(z^2+1)^3[6z^2+8z-2]}{(z+1)^3}$ ...
1
vote
1answer
300 views

Solve $\sqrt{5-12i}$ by square root definition

I KNOW it can be solved by the trig formula, but I want to solve it by the square root definition, so please don't just post an alternative way to do it. By the square root definition: $$z = 5-12i$$ ...
2
votes
1answer
57 views

Derivative complex function

Calculate the derivative of $f(z)=\frac{(1+z^2)^4}{z^2}$ I know that this function is discontinous at $z=0$, what I did is just calculate the derivative the same manner as is done with real functions....
3
votes
0answers
132 views

what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$

If $\frac{a}{a+i}+\frac{b}{b+1}+\frac{c}{c+1}=1$ then what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$ here I got $a=0$ and $bc=1$, when $ bc\neq 0$ but then I cant progress....
0
votes
2answers
68 views

How to prove triangle inequality for euclidean norm on complex number?

We were asked to show that when: $\displaystyle \Vert Z\Vert = \left(\sum_{k=1}^{n} (x_k+iy_k)(x_k-iy_k)\right)^{1/2}$ that $\Vert Z+W\Vert \leq \Vert Z\Vert+\Vert W\Vert$ whenever $Z$ and $W$ are ...
1
vote
1answer
53 views

Limit of product of complex functions, one of which is bounded

Suppose $g$ is a bounded complex function, in other words there exists $M\in \mathbb{R}$ such that $|g(z)|\leq M$, for all $z$. If $\lim_{z \to z_0}f(z)=0$ prove that $\lim_{z \to z_0}f(z)g(z)=0$. ...
1
vote
1answer
202 views

The function $f(z)=|z|^2$ is only differentiable at the origin

Show that a complex function $f(z)=|z|^2$ is continuous on all complex plan $\mathbb{C}$, but it is only differentiable at the origin. I know that a complex function is continuous at $z_0$ $$\...
0
votes
2answers
52 views

the order of zero of rational function

let's say at point $z=a$ we have zero of order $n$ to function $f$. My Question is if there is fast way of knowing the order of zero to function $\dfrac{f}{g}$ in point $a$ where $g(a) \neq 0$ ?