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

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1answer
61 views

Compute imaginary part of the limit

I am trying to compute the imaginary part of this particular question $$\lim_{z\rightarrow i} \frac{iz^3}{z+9 i}$$
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2answers
71 views

Find product of roots of complex equation $z^{12}=-i$

Find product of roots of complex equation $z^{12}=-i$ Let's apply modulus on both sides: $|z^{12}| = |-i|$ It turns out, that $|z|^{12}=1 \implies |z|=1 = r$ Now, use the polar form: $z=re^{i\phi}$ ...
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3answers
101 views

Find sum of roots of complex equation $z^6 = z^{18} = -1$

Find sum of roots of equation $z^6 = z^{18} = -1$ Roots of $z^6=-1$ Roots of $z^{18}=-1$ From the circle I can see, that: $\{z : z^6=-1\} \cap \{z : z^{18}=-1\} = \{z : z^6=-1\}$ So, I can ...
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1answer
45 views

How to find principal argument

How do I find the principal argument of the principal $5$th root of $z$ when $z = 4 + 7i$? I have got the answer as $11$; is that right or I did a mistake?
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1answer
112 views

Problem understanding solution of complex nth-root of unity

a while ago we had the solution for a complex number task about the nth-root of unity in the complex. But now I am having some difficulties to fully understand it: The task was to find all complex ...
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2answers
55 views

Complex Number Problem

I do not know how to do this. 4)a Use de Moivre's to show $\cos 5\theta = \cos \theta ( 16\cos^4\theta -20\cos^2\theta + 5)$ EASY DONE b) Solve $\cos5\theta = 0$ then deduce ...
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1answer
60 views

Weird doubt about complex roots of second grade equation

I'm at the beginning of complex numbers study. I have the following equation: $$ x^2-6i=0$$ It's a second grade eq. so I expect to get two solutions. But: $$x=\sqrt{6}\sqrt[4]{-1}$$So I get 4 ...
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0answers
209 views

intersection of two complex equations

I think the question I'm stuck on is quite simple but for some reason I just can't see how to do it. I'm told that the points A and B represent the complex numbers $z_1$ and $z_2$ respectively. The ...
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1answer
614 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
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2answers
60 views

If $|z_1 - z_2| = |z_1 + z_2|$, then $\arg z_1 - \arg z_2 = \pi/4 $

Problem If $|z_1 - z_2| = |z_1 + z_2|$, show $$\arg(z_1) - \arg(z_2) = \frac{\pi}{4}.$$ Progress I have tried squaring the modulus and using double angle formula for $\tan$, but can't get to the ...
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1answer
83 views

A simple question related to Complex Numbers?

Ok so this was the equation given in my text book $$\implies\sqrt{-a}\sqrt{-a} $$$$= (-1)a $$$$= -a $$ so my question is why can't i solve it this way ...
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1answer
44 views

pseudorandom numbers generation technique and mathematical induction

I have two questions: I am wondering what would be the first $5$ pseudorandom numbers generated by the linear congruential method with modulus $m=7$, multiplier $a=5$, increment $c=2$, and seed ...
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3answers
146 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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2answers
106 views

Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
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0answers
74 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
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2answers
103 views

equations which satisfy argument in complex numbers-help

Please help use an argand diagram to find, in the form a+bi, the complex numbers which satisfy the following pairs of equations. arg(z+1)=1/4π,arg(z-3)=3/4π thanks
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1answer
111 views

How can one find 0?

Suppose one of the three complex numbers, $w_{1},w_{2}$ and $w_{3}$, is $0$. How can one find the one that is $0$ from equations, $$ \left|a_{1,i}w_{1}+a_{2,i}w_{2}+a_{3,i}w_{3}\right|=r_{i}, $$ ...
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2answers
64 views

properties of complex modulus question

If $|a| < 1$, prove that $|z| < 1$ is equivalent to $$\frac{|z - a|}{|1-\bar{a}z|} \leq 1.$$ Where $a$ and $z$ are complex and $\bar{a}$ denotes the conjugate of $a$. I thought this was the ...
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6answers
5k views

How can you find the cubed roots of $i$?

I am trying to figure out what the three possibilities of $z$ are such that $$ z^3=i $$ but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you ...
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2answers
43 views

ODE with complex char roots gives strange solutions

$y''-4y'+5y=0$ has char roots - $\{e^{(2+i)x},e^{(2-i)x}\}.$ So its solutions is $e^{2x}\cos(x), e^{2x}\sin(x).$ But when i plug, e.g., first of them into original eq. i get: $-4 e^{2x} cos(x) + 8 ...
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1answer
400 views

i^i^i^i^… Is there a pattern? [duplicate]

I was messing around with $i$ and I (haha) noticed that certain progressions arise when I keep on raising $i$ to $i$ to $i$ and so forth. Though, I am not really quite sure what is going on (and I ...
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2answers
169 views

Difference between i and -i

Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a - bi$ does the image ...
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4answers
141 views

Find $(a+ib)^{492}$ given that $(a+ib)^{493}=1$ [closed]

We are given that: $(a+ib)^{493}=1$ . Find the possible values of $(a+ib)^{492}$.
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2answers
73 views

Find radius of convergence of power series

Since we know that given $\sum_{n=0}^{\infty }C_nz^n$, if $\lim_{n\rightarrow \infty }|C_n|^{1/n}$ exists then $R^{-1}=\lim_{n\rightarrow \infty }|C_n|^{1/n}$ where $R$ is the radius of convergence. ...
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6answers
1k views

Applications of Complex Numbers

For my Complex Analysis course, we are to look up applications of Complex Numbers in the real world. The semester has just started and I am still new to the complex field. I want to get a head start ...
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1answer
79 views

What happens if I remove the sign from the exponential of Fourier Transform?

Forward Fourier Transform $\hat f(x)$ is defined such as $$ \hat f(x) = \int_{-\infty}^{\infty}{f(t) e^{-2\pi t x i} dt} $$ but I am wondering what happens if I define it $$ \hat f(x) = ...
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2answers
66 views

Evaluate equation $z^{13}=\overline{z}$

I need to find all solutions of such complex number equation: $z^{13}=\overline{z}$ We assume that $r = |z|$ and we use Euler's formula $z=|z|e^{i\phi}=re^{i\phi}$. Then, multiplying both sides by ...
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1answer
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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0answers
53 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
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0answers
19 views

Alternative coordinates for the complex plane $\mathrm{Re}[e^{-is}z]=a$, $\mathrm{Re}[e^{-it}z]=b $

I am defining coordintes on $\mathbb{C}$ using a "generalized" real and imaginary part. Here $a,b \in \mathbb{R}$. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ...
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1answer
71 views

Complex Exponential/Trigonometric Functions

I'm having trouble on proving the following state of a Lemma using the power series of $\exp z$ centered at $0$: For all $z \in C$: $\exp(z + 2\pi i) = \exp(z)$ and $\exp(z) \neq 0$ All help ...
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1answer
72 views

If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is

If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is $\bf{My\; Try::}$ Given $11z^{10}+10iz^9+10iz-11 = 0\Rightarrow \displaystyle z^9 = \frac{11-10iz}{11z+10i}.$ Now Put $z = ...
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1answer
136 views

How do I solve $|\sinh(x+iy)|^2 = (\sin(y))^2+(\sinh(x))^2$

How do I solve this ? $|\sinh (x+iy)|^2 = ( \sin (y))^2+ ( \sinh (x))^2$ I'm not sure how to solve the left hand side.
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3answers
194 views

let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant

$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant... Then we are supposed to use this result to ...
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2answers
74 views

Argument of a complex number

Find the argument $ \theta $ of a complex number z that satisfies the following condition: $|exp (z^3)| \to 0$ as $|z| \to \infty$ I suspect the real part of $z^3$ must be negative and this is enough ...
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3answers
184 views

Radius of convergence of the series $\displaystyle\sum\limits_{n=0}^\infty \frac{n!\,z^{2n}}{(1+n^2)^n}$

I am doing the following problem and would like to know whether my answer is correct or not: Find the Radius of convergence for the complex series $\displaystyle\sum\limits_{n=1}^n ...
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1answer
38 views

How is this computed?

$ \left| 1-e^{-is\lambda} \right|^2 = 2 (1-cos\lambda s)$ where $i=\sqrt{-1}$ I don't know how to work with $i$. Thanks
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1answer
49 views

Graphing $\sqrt { -x}$

how does my calculator graph ($\sqrt { -x}$. Since I can't graph a complex number, how does my calculator graph the $\sqrt { -x}$ ?
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1answer
114 views

From 'The Joy of x' book: John Hubbard and problems with multiple roots

My math skills are super rusty. In an effort to get some vigor back I started some reading and picked up The Joy of x based on its rave reviews.. I just couldn't make any sense out of the following ...
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7answers
614 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
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2answers
113 views

Results Analogous to the Two and Four Square Theorems.

A result that arises out of the study of $\mathbb{Z}[i]$ is that the following are equivalent for integer primes p: 1) $p\equiv 1$ (mod 4) or $p=2$ 2) $\exists a,b\in\mathbb{Z}$ such that ...
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1answer
110 views

Use algebra of Big-O notation to express tan($z$)

We can use the definition of Big-O notation to simply prove that $\sin(z)=z-\frac{z^3}{6}+O(z^5)$ as $z\rightarrow 0$, $\cos(z)=1-\frac{z^2}{2}+O(z^4)$ as $z\rightarrow 0$ and ...
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2answers
60 views

Proving an inequality with the Schwarz inequality

Given a vector space with a Hermitian dot product defined, prove the following inequality using the Schwarz inequality. Let $f$ be a complex value function that is continuous within $0 \le x \le 1$, ...
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2answers
85 views

Complex Numbers - Polar Form/Algebraic Form

I am having some problem with this question: Write the following complex number's in the algebraic form: $\dfrac{5i}{(1-2i)(1-i)(1+3i)}$
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2answers
90 views

Prove that 1 has n distinct roots of order n

I am trying to show that 1 has n distinct roots of degree n, or in other word that the equations $$z^n=1$$ has n different roots over the complex field. I know that the fundamental theorem of Algebra ...
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1answer
124 views

Dilations - Complex Numbers

Not really sure where to start...hints are appreciated, thanks. In this problem, we will show that the composition of two dilations is, in general, another dilation. (a) Let $ z_0$ be an arbitrary ...
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2answers
328 views

Complex Numbers Geometry

I'm not sure where to begin on this problem - do I plug in for a and solve for z? I was also given a hint: Let z be a point on the line we're trying to describe. We have good tools in complex ...
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2answers
119 views

Solve the equation $x^{2n} + 1 = 0.$ Use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients.

I am asked to solve the equation $x^{2n} + 1 = 0,$ and to use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients. I am given the hint that pairing factors arising from ...
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3answers
39 views

Value of the complex expression

How can I calculate the exact value of something like that: $|e^{\sqrt{i}}|$
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3answers
109 views

Determine all complex number for which: $ \arg(Z^6) = \arg(-Z^2),\ \mathrm{Re}(Z^3) = 2 $

While preparing for the next semester, I stumbled upon this complex number problem which kind of confuses me. I know it has something to do with this - but I simply ...