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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50 views

What are the properties of the fourier transform of a phase-only function?

Given a function of the form: $$ f(x) = e^{i\phi(x)} | \phi(x)\in\Re $$ What are the properties of its Fourier transform? For instance, purely real functions have Fourier transforms with symmetric ...
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2answers
46 views

finding all the roots (including complex)of the equation

Find all the roots of $z^4=16(z+2i)^4$. Can someone help me teach/ guide to solve this equation?
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0answers
25 views

For wich $u\in\mathbb{C}$ is $0^{u}$ defined?

It is obvious that $\left|e^{v}\right|=e^{\text{Re }v}>0$ showing that $\ln z$ is not defined for $z=0$ . So the expression $z^{u}=e^{u\ln z}$ cannot be used here. Nevertheless we don't hesitate ...
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2answers
159 views

Prove that $f$ is constant if $f$ is real when $|z|=1$ [duplicate]

Let $f$ be a holomorphic function in $\mathbb{C}$. Prove that if $f$ is real when $|z|=1$, then $f$ must be a constant function. I honestly do not know how to do this problem, consider using ...
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2answers
45 views

Make the vector $[1,1]$ turn of an angle - $\pi/4$ , with complex numbers

We have $[1,1]$ and $\theta = -\pi/4$ here is my attempt: $(\cos(-\pi/4) + i \sin(-\pi/4)) * (x+iy)$ = $(\sqrt{2}/2 - i \sqrt{2}/2) (1+i)$ = $\sqrt{2}/2 - i^2\sqrt{2}/2 $ = $[\sqrt{2}/2 + ...
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1answer
64 views

Show that $f=g$, if $f(z)=g(z)$ for $z\in dA$ with $A$ bounded region

Let $A$ be a bounded region, $f$, $g$ continuous functions of $\bar{A}$ in the complex. Suppose that these functions are holomorphic in the region and agree on the border. Prove you are the same. I ...
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2answers
47 views

Geometric position of a complex set.

I'm quite struggling with this question and help would be appreciated. Define the geometric position of a complex set: $|z-4i| +|z+4i|=10$
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2answers
68 views

Finding the roots of 4096x^3-10496x^2+152576x - 961=0 (1 root and 2 complex)?

I don't know how to find the roots of 4096x^3-10496x^2+152576x - 961=0 I try using wolfram and http://en.wikipedia.org/wiki/Cubic_function. I don't really understand it can someone please explain how ...
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4answers
94 views

Is posible proof that there is not a holomorphic function $f:\Delta\rightarrow\bar{\Delta}$, such that $f(0)=0$ and $f(z)=i$ for some $z\in\Delta$.

I have problems with this: Proof that there is not a holomorphic function $f:\Delta\rightarrow\bar{\Delta}$, such that $f(0)=0$ and $f(z)=i$ for some $z\in\Delta$. My problem is that I am getting ...
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1answer
70 views

Show that $\frac{|F(z)-F(a)|}{|F(z)-\bar{F(a)}|}\le\frac{|z-a|}{|z-\bar{a}|}$ if $z\in\Pi^{+}=\{z\in\mathbb{C}:Im(z)>0\}$

Consider $\Pi^{+}=\{z\in\mathbb{C}:Im(z)>0\}$ and let $a\in\Pi^{+}$. Suppose that $F:\Pi^{+} \rightarrow \Pi^{+}$ is holomorphic. Prove that for all $z\in\Pi^{+}$ we have: ...
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1answer
18 views

Step in finding $\sin^{-1}z = w$ for a fixed complex $w$ and unknown complex $z$

This is in the section of the book preceding a general formula but I don't know how the author arrives to the second equation in the picture. The closes I have gotten to it is $$2iz = ...
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3answers
65 views

Complex number: Roots

Solve all the roots of the following equation: $$(z-i)^2(z+i)^2=\frac{1}{4}.$$ Find the set of complex numbers $z$ such that $$\left|\frac{z-3}{z+3}\right|=2.$$ Would anyone mind telling me how ...
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1answer
307 views

Complex Numbers and Transformations

If a transformation t acts by rotating every point of the plane around the origin by $\pi/5$ clockwise and then proceeds to translate it by vector $v$ = $(1,2)$. How do I describe this ...
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2answers
56 views

Are complex conjugates unique?

I'm trying to decide if a function $\varphi:\mathbb{C}\rightarrow \mathbb{C}$ defined by $\varphi(\alpha)=\bar{\alpha}$ is onto or one to one. I know it will be onto because every element in ...
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1answer
97 views

Simplification of a trilogarithm of a complex argument

Is it possible to simplify the following expression? $$\large\Im\,\operatorname{Li}_3\left(-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}\right)$$ where ...
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1answer
138 views

Does this follow from the definition of the LambertW function?

The LambertW function $W(s)$ also called ProductLog seems to satisfy this relation: $$-W(s) = \underbrace{-s e^{-s e^{\cdot^{\cdot^{-s e^{-W(s)}}}}}}_n$$ Or truncated: $$-W(s) = -se^{-s e^{-s e^{-s ...
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1answer
306 views

Help with rearranging equation to get real and imaginary parts..

I know this is so simple but my algebra is totally failing me.. I have the equation 1/1+2i and I want to extract the real and imaginary parts so I have it in the form.. Re+Im could someone just ...
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1answer
154 views

How should I the Residue Theorem to evaluate the integral $\int_{|z|=2} \frac{dz}{(z − 4)(z^3 − 1)}$?

How should I use the Residue Theorem to evaluate the integral $$ \int_{|z|=2}\frac{dz}{(z − 4)(z^3 − 1)}?$$
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2answers
287 views

What does $i^i $ equal and why? [duplicate]

I've been reading up on why the value of 0^0 is controversial (see Zero to the zero power - Is $0^0=1$?) and I wondered: is it possible for $i^i$ to have a value? I plugged it into a TI-83 calculator ...
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1answer
128 views

What is the value of $\int_{-\infty}^{\infty} \frac{e^{-ix}}{x^{2 }+ 4} dx$ [duplicate]

What is the value of: $$ \int_{-\infty}^{\infty} \frac{e^{-ix}}{x^{2}+ 4} dx $$ And also maybe the problem should be: $$ \int_{-\infty}^{\infty} \frac{e^{-3ix}}{x^{2}+ 4} dx ...
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1answer
60 views

Some identity with a complex function

Let $f$ be a complex function. I've got the following equation to prove: $$I_f(z) = |\frac{df}{dz}|^2 - |\frac{df}{d\bar{z}}|^2 $$, where $I$ is determinant of Jacobian matrix. But knowing that $ ...
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1answer
166 views

Axiomatic definition of complex numbers

Trying to build axiomatically the set $\mathbb C$ of complex numbers, my first attempt was to define $\mathbb C$ with three structures: addition, multiplication and conjugate: $\langle\mathbb ...
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1answer
80 views

Laurent series of an analytic function divided by $z$

This is a probably basic question about Laurent series. Say $g(z)$ is an analytic function, that $g(0) = 0$, and $f(z) = g(z)/z$. My textbook says $z = 0$ is a removable singularity of $f(z)$. A ...
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2answers
63 views

The calculation of roots of complex numbers.

How to calculate the roots of $x^6+64=0$? Or how to calculate the roots of $1+x^{2n}=0$? Give its easy and understanble solution method. Thank you. In general, the results of "exp" are obtained.
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1answer
40 views

$-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?

Given $a>0$ I would like to know whether: $\alpha=-ia(1+ \sqrt{1-1/a^2})$ and $\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle. How can I check that?
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1answer
487 views

Find solutions for $z^3 + 4\sqrt{3} -4i = 0$ ($n$th roots of complex numbers)

I'm not sure if I have done the last parts of this right. $z^3=-4\sqrt3+4i$ Let $z^3=w$ So $w=-4\sqrt3+4i = 0$ $|w| = \sqrt{-4\sqrt3^2+4^2}$ $=8$ I have drawn the points $4$ and $-4\sqrt3$ on a ...
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34 views

Residues/Power Series

Use the fact $e^z = -1$ when $z = (2n+1) \pi \imath$ to show that $e^{1/z}$ assumes the values $-1$ an infinite number of times in each neighborhood of the origin. More precisely, show that $e^{1/z} = ...
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1answer
37 views

Find the minimum value of $\operatorname{Im}(z^5)/(\operatorname{Im}(z))^5$ [closed]

If $z$ is a complex number, then find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5},$$ where $\operatorname{Im}(z)$ denotes the imaginary part of z.
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0answers
116 views

Development of imaginary exponent without appealing to “ambiguity” between $i$ and $-i$

Is there a way to develop the definition of the imaginary exponent, $z^i$, for complex $z$, that does not appeal to the notion that $i$ and $-i$ are "qualitatively indistinct" and that does not rely ...
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1answer
239 views

Finding modulus and argument of z³ - 4√3 + 4i = 0

I think I am messing up somewhere as the principle argument should be a nice number from the standard triangles such as $\fracπ4$, $\fracπ3$ or $\fracπ6$ or something close. (That's what we have ...
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1answer
134 views

What line are inverse functions on the complex plane reflected over?

On the real plane (xy plane) inverse functions are reflections of their original functions over y=x. Is there such line for complex functions and their inverses?
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0answers
26 views

Numbers $e^{iz}$, where $\Re(z) \in (0, \frac{\pi}{2})$

What part of complex plane are numbers $e^{iz}$, where $\Re(z) \in (0, \frac{\pi}{2})$? Can you help? I've been trying to figure that out for quite a long time and still don't know...
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2answers
97 views

Can I approximate a complex number by its imaginary part, if real part is small compared to imaginary part?

I have the following doubt. How do you explain this? Here $j$ means $\sqrt{-1}$.
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1answer
177 views

Inversion applied on circles

I'm studying for my exam and one of the questions I am stuck on is: Show that under inversion in the unit circle a circle with centre C and radius $S$ inverts into a circle with centre ...
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1answer
177 views

Geometry of Complex Numbers

Write down in the form ${Z}\rightarrow{AZ+B}$ the following transformations of the complex plane: (a) translation in the direction $(2,-3)$ (b) rotation about (0,1) through $\pi/4$ I know from my ...
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1answer
93 views

Finding all roots of complex equation with z

Hi I would like to find the all the roots of the equation $$e^{j\pi/3} z^5 + 4 e^{5j \pi /6} z^3 + z^2 + 4j =0$$
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2answers
81 views

Complex integral of an exponent divided by a linear ($\int \frac{e^u}{u-1}$)

Here is the question I'm working on: Evaluate the following integral: $$ \oint_{|z+1|=1} \frac{\sin \frac{\pi z}{4}}{z^2-1}dz$$ I've tried along the following line. Substitute $sin(z) = ...
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0answers
100 views

Part of an unit circle $e^{iz}$, where $\Re(z) \in (0, \frac{\pi}{2})$

I have a unit circle $e^{iz}$ . Let's think about such $z \in C$, that $\Re(z) \in (0, \frac{\pi}{2})$. Which part of my unit circle is this? How to find the answer? I know that $z$ is the angle ...
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2answers
60 views

Evaluating $\frac{(1+i)^{n+2}}{(1-i)^n}$

Evaluate $\dfrac{(1+i)^{n+2}}{(1-i)^n}$ I think that the meaning is that it need to be simplified. Thanks
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0answers
166 views

Correct way to write the polar form of a complex number

What is the most correct way to write the polar form of a complex number? For example, given the complex number: $\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i$ I would write the polar form as follows: ...
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1answer
159 views

Book suggestion- complex analysis -conformal mapping.

I am studying complex analysis. And I am using J. Bak and D.J. Newman's book.(springer) And now my studying topic is conformal map. In addition to this book, I want to learn other book names which ...
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2answers
124 views

WolframAlpha seems to suggest that $log(z) = log(-z)$

Wolfram-Alpha returns this snippet at the end of the "step by step solution": I'm confused by the negation of the expression inside the logarithm. The definition of complex logarithm is: $$log(z) ...
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1answer
124 views

History of complex numbers

I'm interested in the history of complex numbers - their origin and their subsequent development. I'd be very interested if anyone can provide references for finding out more about this topic.
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2answers
823 views

Find all the values of $(1+i)^{(1-i)}$

The question says to find all the values of $(1+i)^{(1-i)}$ I have trouble figuring out firstly, exactly what values are being looked for. I can toy around with the equation a bit to try to make it ...
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1answer
75 views

Modulus of solutions to fourth-degree equation

The equation $z^4-6z+3=0$ has four complex solutions. How many of them satisfy $1<|z|<2$? I am trying to apply Rouche's theorem. On the boundary $|z|=2$, we have $|-6z+3|\leq 15<16=|z^4|$, ...
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2answers
155 views

Rewriting $x^3-3xy^2+2xy+i(-y^3+3x^2y-x^2+y^2 )$ in terms of $z$, with $z=x+yi$

How do I write $f=u+iv$ with: $u=x^3-3xy^2+2xy$ and $v=-y^3+3x^2y-x^2+y^2 $ in terms of $z$ with $z=x+yi$?
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1answer
52 views

How to show that all roots of $(11+v)q^3-18q^2+9q-2$ have their absolute value less than 1.

The equation is $(11+v)q^3-18q^2+9q-2=0$, where $v>0$ I need to show that either absolute value of all the roots is not greater than one or there exists a root $q: |q|>1$. Using Weierstrass ...
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2answers
62 views

Set in the Complex Plane

How can I describe the set: $$ \left\vert z - {\rm i}\,\right\vert = 3\left\vert z\right\vert $$ It does appear quite unfamiliar. Attempt: $$ \left\vert\frac{z-i}{z}\right\vert = 3 $$ so, $$ ...
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1answer
110 views

Compute $\int c Re(z) dz$ where $c$ is the directed line segment from $z = 0$ to $z = 1 + 2i$

first we see that $x(t) = t$ where $0<= t <= 1$ and $y(t) = 2t$ where $0<= t <=1$ and so $$z(t) = t(1+2i) $$ but then $Re(z)$ how will I compute that ?
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3answers
811 views

Find all complex solutions of $\sin(z)=1$ [closed]

Find all complex solutions of $\sin(z)=1$. How would I go about this?