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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46 views

Finding the set of analytic functions whose image is a subset of a given set

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help ...
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75 views

Question about a step in the proof of the Cauchy-Schwarz inequality in $\mathbb{C}$

I'm studying the proof of the Cauchy-Schwarz inequality, which states that for complex numbers $z_1,\ldots. z_n,w_1,\ldots, w_n$ we have $$ \Big\vert\sum_{j=1}^nz_jw_j \Big\vert^2\le \sum_{j=1}^n\vert ...
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46 views

$D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact

This is the proof I wrote for $D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact. $$ \bigcup_{n=2}^{\infty}D_{1-(1/n)}(0) $$ is clearly a open covering of $D_1(0)$. Consider the finite ...
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27 views

The annulus $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open

I want to prove that the set $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open. This is my attempt. Let $z \in A_{r,s}(z_0)$. Then $|z-z_0|-r>0$. Let $r'=[|z-z_0|-r]/2$. Then ...
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51 views

Third point of a triangle in the complex plane

I have an equilateral triangle with two points equal to $(2+2i)$ and $(5+i)$. I want to find the third point(s) (there are $2$ of these). I have that the side length of the triangle is $\sqrt{10}$.
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84 views

Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?

Is there anything special with the form: $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers. In fact, I saw it in a problem: If $|z| = 1$, prove that ...
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47 views

Finding $\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$

$\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$ I am having trouble getting a final answer that makes sense to me. Here is what I tried: ...
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32 views

Help with proof in complex analysis

I was looking at the proof of the result, the image of $\mathbb R_\infty$ under mobius transformation is a circle. I don't follow how does this step $(a\bar d-\bar bc)w+(b\bar c-\bar ad)\bar w+b\bar ...
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31 views

Number of complex numbers such that $z^{80} = 1$ and other properties

Let $$A = \left \{ z \in \mathbb{C} : \Re z > 0, \Im z < 0, z^{80} = 1 \right \}$$ Then, the number of elements in $A$ is $19$, $20$, $21$, or $22$? I just started studying complex ...
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480 views

Sketching regions is complex plane

When sektching the region $\left|\frac{2z-1}{z+i}\right|$$\geq$1 on the argrand diagram, how should we go about identifying the region, should we take $\left|2z-1\right|\geq\left|z+i\right|$ or ...
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66 views

Show a complex equation has one or two roots

Let $a$ $\neq$ $0$, $b,$ and $c$ be complex constants. Show that the quadratic equation $az^2+bz+c=0$ has one or two roots. My thoughts: Let $a=a_1+ia_2,$ $b=b_1+ib_2,$ and $c=c_1+ic_2$. I also ...
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456 views

Find the cube roots of $ -8 i $ and plot them on a plane.

I can’t figure out the angle of this equation. I set it up like this: $$ z^{3} = 0 - 8 i. $$ I find that the $ r $-value is $ 2 $, but when I try to find the angle, I’m stuck. I can’t divide by $ 0 ...
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27 views

Images of the stereographic projection's inverse

I am trying to solve a problem which states: Let $\phi: \bar{\mathbb C} \to S^2$ be the inverse function of the stereographic projection Calculate $\phi(Re(z))=0$ and $\phi(Im(z))=0$. I can guess ...
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105 views

What is the real and imaginary parts for the complex function $f(z)=z^z$

I know: $f(z)=z^z =|z|^ze^{iz\theta} $ and $=|z|^z(\cos(zθ) + i\sin(zθ))$ But how do I continue to get the results for $\Re(z^z)$ and $\Im(z^z)$? $$\text { }$$ Thanks.
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79 views

Can there be a complex line?

In an early math class, I was shown how all Reals could be constructed from Rationals using a 2-D representation (ex. Real numbers are represented by (a + b \sqrt{2} ) where a & b are Rational). ...
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35 views

Query regarding Linear Transformation…

As we always read in Complex Analysis, Linear Transformation (L.T.) is a combination of Translation, Rotation and Magnification i.e. $T(z)=az+b$ is a L.T. in complex. However, It doesn't satisfy the ...
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80 views

Construct a non-constant analytic function $f : \Omega_1 \to \Omega_2$ or show that this is impossible.

I am having a lot of difficulty with the following past qualifying exam problem. Any help would be awesome. Thanks. Let $\Omega_1 = \mathbb{C}\setminus \left \{\{0\} \cup \{\dfrac{1}{n}:n\in \Bbb ...
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76 views

Solve $(\frac{z+1}{z})^5 =1$ using fifth roots of unity

$$(\frac{z+1}{z})^5=1$$ Show that its roots are $$-\frac{1}{2}(1+i\cot(\frac{kπ}{5})), k = 1,2,3,4$$ I need to use the five fifth roots of unit, with angles $0,\frac{π}{5}, ...
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68 views

What does $\theta = \text{arg}(a,b)$ mean?

I have this equation where an angle is calculated using following formula: $$\theta = \text{arg}(C_1, C_2)$$ where $C_1, C_2$ are some numerical values. What exactly does it mean?
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248 views

Complex Analysis: Isolated Singularities, Poles, and Residues

I was given the following question. Show that the isolated singularities of the function $f(z) = \frac{z}{z^4+4}$ are poles. Determine the order of each pole and find the corresponding ...
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201 views

Complex Analysis: Cauchy Integral Formula

I came across this problem. Let $C$ denote the circle {|z| =2}, parametrized as a positively oriented simple closed curve. Evaluate $\int_c\frac{1}{z^2-1}dz$ I want to approach this ...
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100 views

Tricky Argand diagram sketch

I'm trying to sketch $D=\{z \in \mathbb{C}: 2 \leq \vert z \vert < \vert z-2 \vert <4\}$. I know that, geometrically, this is all $z$ whose distance from the origin is $\geq 2$ and is $<$ ...
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54 views

Complex Logarithm Derivation

I don't understand how the definition of the complex logarithm was derived. It is $ log(z) = ln|z| + i Arg (z) $, where $ z = x + iy $. I've tried all sorts of method to find this definition but ...
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22 views

$|g(t_{1}) e^{-(t_{1}-x)^{2}}- g(t_{2})e^{-(t_{2}-x)^{2}}|\leq |f(t_{1}) e^{-(t_{1}-x)^{2}}- f(t_{2})e^{-(t_{2}-x)^{2}}| $?

Suppose $f, g: \mathbb R \to \mathbb C$ such that $|g(t_{1}) -g(t_{2})| \leq |f(t_{1})- f(t_{2})| $ for every $t_{1}, t_{2} \in \mathbb R.$ Take any $x\in \mathbb R$ and fix it. Edit: We also assume ...
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145 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
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123 views

Clarification on a step in the proof of Lagrange's identity for complex numbers.

I wrote this proof of the following identity and I want to verify that a certain step is correct. $\newcommand{\conj}[1]{\overline{\vphantom{b}#1}}$ $\newcommand{\on}[1]{\operatorname{#1}}$ ...
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126 views

Finding the locus represented by complex variable equations?

I'm trying to solve these two problems related to complex number but hardly found a solution. I hope that someone can solve these and clear it up for me. Thank you. |z+2|=2|z-1| |z+5|-|z-5|=6
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28 views

A complex metric

Given the following definition $d(z , w) = \begin{cases}0 & z=w \\ |z|+ |w| & z\neq w \end{cases}$ I have to prove that $d(z,w)= 0\Rightarrow z = w$ Which is in part of checking that $d$ is ...
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Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
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36 views

Complex Transformation

$z_1 = 1 + i$ and $z_2 = -1 + i$ I am told: $w = \dfrac{az + b}{z + d}$ where $z \not= -d$ Where a, b and d are complex numbers, maps the complex number $z$ onto the complex number $w$. Given that ...
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133 views

Using Liouvilles theorem to show that f is identically constant on all of $\mathbb C$

Use Liouvilles theorem and the fundamental theorem of calculus to prove that for an entire function $f$, if there exists $M \in \mathbb R: Re(f(z)) \leq M$ $ \forall z \in \mathbb C $, then $f$ is ...
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94 views

General formula for a square in complex numbers

I need to find a general formulae for a square, with its interior included, in terms of complex numbers. Note that your general square should have (general centre, side-length and orientation.) I do ...
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250 views

How to find number of real and complex roots?

Below is a question asked in JNU Entrance exam for M.Tech/PhD. I want to know if there is a fixed way to calculate it. I have failed to use the factor theorem. ...
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62 views

showing that $(z_1^2z_2)^2$ is a real number

Given $z_1=a+bi,z_2=c+di,\frac{b}{a}=\frac{d}{c}=\frac{1}{\sqrt3}$, $a,b,c,d$ are real numbers; $z_1,z_2$ are complex numbers. Need to prove that $(z_1^2z_2)^2$ is a real number. So i figured that ...
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50 views

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$?

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$? Meaning, if I have a complex number and I multiply it by its complex conjugate, would that always return a number in $\mathbb{R}$? ...
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61 views

Need help solving this equation with complex numbers

$$(z^3 + 1)^3 = 1$$ where $z$ is an element of the complex number system. Can someone show me the most efficient way of finding all the solutions for $z$ here and also if possible please demonstrate ...
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134 views

Integral calculus use of Newton-Leibnitz rule

My friend asked me this question: If $y(x)= \int_{0}^{x}f(t)\sin{(px-pt)}dt$ then what is the value of $y''(x)-((p^2)*y(x))$. He gave me the hint to consider $\sin(px-pt)$ as the imaginary part of ...
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39 views

How to find polar values of complex number as quick as possible?

I need to calculate these kind of values in exams in best speedy way. Convert $1.46 + 3.17j$ to polar form ($r∠θ$) Is there is any solution to find of the values as quick as possible? By the way, ...
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99 views

Find the circle equation passing through i, 1+i, 2-i in the form of |z-p| = r|z-q|

I know that the circle equation becomes $|w - c/1-r^2| = |c| \frac r{1-r^2}$. I started by assigning $i$, $1+i$, $2-i$ to the variables $z$, $p$, and $q$ respectively. I am not sure I am doing this ...
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118 views

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$.

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$. I have already proved that other functions are continuous by using that $f, g$ are continuous implies $f+g$ and $fg$ are continuous. ...
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90 views

complex conjugate

Prove that $|z+w| \leq |z|+|w|$. Okay so I labeled $z$ as $z=a+bi$ and $w$ as $w=c+di$. Substituting and solving for everything I got $\sqrt{(a+c)^2 + (b+d)^2} \leq \sqrt{a^2 +b^2} + \sqrt{c^2 ...
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162 views

How to calculate $\theta$ when we know $\tan \theta$.

Hej I'm having difficulties calculating the angle given the tangent. Example: In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this ...
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42 views

Laurent Series / Residue Theorem

I'm having trouble on computing $\int_\gamma \frac{dz}{(z^2-4)(z-2)}$, where $\gamma$ is the positively oriented circle centered at 2 of radius 1. Any help on this will be very appreciated.
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42 views

Is convolution associative with regards to the complex unity?

Setup: I need to do a convolution with the function $\cfrac{i}{x}$, and I would like to get rid of the $i$. My functions to be convolved are all real valued. According to the ever-failable wikipedia, ...
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72 views

Complex root won't work

So I'm trying to get this: http://www.wolframalpha.com/input/?i=%288*sqrt%283%29%29%2F%28z%5E4%2B8%29%3Di And I've calculated $z^4=16 \left( \cos (\frac{- \pi}{3})+ \sin ( \frac{- \pi}{3}) \right)$ ...
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92 views

How to solve $(1+i\sqrt{3})^{-1+i}$??

Good morning, I want to solve this... but I lose my way. I hope somebody help me... I show you my calculus $(1+i\sqrt{3})^{-1+i}=e^{(-1+i)\log(-1+i)}$ $(1+i\sqrt{3})^{-1+i}=e^{(-1+i)(\log ...
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82 views

complex numbers, complex roots of equation.

$z_1=a+bi$ , $a,b\in\Bbb R$, $b\neq 0$ is a complex root of the equation $z^2-2z+25=0$. Without evaluating the roots, answer the following questions: i) show that $\overline{z_1}$, the conjugate of ...
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64 views

Conversion of complex numbers to standard form

$$(1/\sqrt{2} - i/\sqrt{2})^8$$ Im not sure where to begin here, should i just expand it out completely and then simplify? $$1/i^{2013}$$ For this one im guessing because $2012$ is basically the same ...
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68 views

Find the value of K. Use of l Hospital's rule and expansion is not allowed.

Let $f(x) =\log_{cos3x} (\cos2ix)$ if $x \ne 0$ and $f(0) = k$ where ($i$= iota) is continuous at $x = 0$, then find the value of $K$. Use of l Hospital's rule and expansion is not allowed.
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90 views

Calculus $T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$

Calculus: $$T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$$ and $$T_2=\prod_{k=1}^{n-1}\sin\frac{k\pi}{2n}$$ My tried: I use Euler's formal: $$z_k=e^{i\frac{k\pi}{2n}}=\cos\frac{k\pi}{2n}+i\sin ...