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

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1answer
50 views

Evaluate $\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos (\theta )\right)^2}$

I am trying to evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos \left(\theta\right)\right)^2}$$ via change of variables and applying Cauchy's Residue Theorem. Here is how I'm ...
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1answer
11 views

Limit of complex exponential

The following is the characteristic function of a random variable $X_n$:$$\phi_{n}(t)=\frac{1-e^{it}e^{\frac{it}{n}}}{(n+1)\left(1-e^{\frac{it}{n}}\right)}$$ for $t \in \mathbb R$. I am trying to ...
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2answers
43 views

Faster way to for $z^3 = -2 (1+i \sqrt 3) \bar z$ than complex algebra

What is the fastest way to solve for $z^3 = -2 (1+i \sqrt 3) \bar z$? I know how to do this using complex algebra. but that takes a long time. Can someone show me a faster way?
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0answers
57 views

What is a complex number that can't be written in polar form?

What is the cartesian form of a complex number that can't be written in polar form? Why can't it be written in polar form?
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1answer
19 views

Prove of an addition theorem for the general binomial coefficients

Prove that: $\sum_{k=0}^n \binom{s}{k} \binom{t}{n - k} = \binom{s + t}{n}$ for all $s, t \in\Bbb C $, $n \in N\cup {0}$. That's pretty much all I'm given, and therefore, I haven't come quite far ...
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0answers
16 views

How to determine the limit of a complex function

It is easy to show that a complex function doesn't have a limit as it approaches a certain point, but is there any way to know for sure whether any given complex function has a limit as it approaches ...
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1answer
24 views

Circle in the plane of complex numbers

Let $K = \{z \in \mathbb{C}: |z−a|=r \}$ be a circle in $ℂ$. Show that, for the case that $|a|$ is not equal to r, the image of $K$ under the transformation $z$ $\to$ $\frac {1}{z}$ is a circle too. ...
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1answer
30 views

Relating convergence theorem for Newton-Raphson method to Newton fractal

I have created a Newton fractal (below) using the Newton-Raphson method to find the five solutions of f = (z^5-1) The convergence theorem of Newtons method say ...
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3answers
41 views

Finding the roots of a polynomial on a complex plane [duplicate]

I use an online calculator in order to calculate $x^5-1=0$ I get the results x1=1 x2=0.30902+0.95106∗i x3=0.30902−0.95106∗i x4=−0.80902+0.58779∗i x5=−0.80902−0.58779∗i I know that this is the ...
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1answer
41 views

complex plane questions

Find where the points of the complex plane are if, a) |pi - arg z| < pi/4 b) |Re z| < 1 c) Im {(z+1)/(z+i)} = 0 d) z = z1 + t(cosx + isinx), 0<=x<=pi/4 where z1 = 1+2i and t=2 Please ...
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3answers
52 views

Factoring a polynomial of fourth degree with false roots: $x^4+4$

I want to write this polynomial in factored form: $$x^4+4$$ The reason I want to do this is to be able to make partial-fraction decomposition on it to make an integrand easier to integrate. What's ...
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1answer
13 views

A linear fractional transformation and mapping of concentric circles

Q: A fractional linear transformation maps the annulus $r < \|z \| <1$ (where $r > 0$) onto the domain bounded by the two circles $\|z- \frac{1}{4} \|=\frac{1}{4}$ and $\|z \|=1$. Find $r$. ...
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1answer
26 views

Where the power series is convergent

Where $f(z)=\sum_{n=1}^{\infty}\frac{(2i)^n}{n}z^n$ is convergent? I checked that the radius of convergence is equal to $\frac{1}{2}$. Now, since we know that the series ...
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0answers
26 views

A function on the punctured complex plane which turns out to be constant

Let $f: \mathbb{C}- \{0\} \rightarrow \mathbb{C}$ be a holomorphic function on the punctured complex plane, and suppose that $f(2z)=f(x)$ for all $z \neq 0$. Prove that $f$ is constant. Proof: ...
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1answer
23 views

Laurent Series Expansion for $f(z)=\dfrac{z+2}{(z+1)(z-2)}$ in $\{1<|z|<2\}$ and $\{2<|z|<\infty\}$

I'm trying to get the Laurent Series expansion of the function stated in the title in the stated regions. My approach is as follows: We can first break up $f(z)$ using partial fractions ...
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1answer
30 views

Zeros of $z^2 \text{cos}z^2$

Is there an easy way to find the zeros of the function $z^2 \text{cos}z^2$, $z\in \mathbb{C}$ and the respective orders (multiplicities)? All I can think of is to find $f^{(1)},f^{(2)},...$ but then ...
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1answer
19 views

Find the basis of set given by matrices

In linear space of matrix $2\times 3$ over $C$ we have subspace generated by: $ A= \{{\left[\begin{array}{ccc}i&i&i\\i&0&1\end{array}\right]}$ ...
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1answer
13 views

Phase of relative coordinate in the complex plane

If we have two points $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ in the complex plane and define the relative coordinate $z=z_2-z_1$, we have that the length of $z$ is the Euclidian distance between the ...
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1answer
39 views

Solve $z^{1+i}=4$

$\def\Log{\operatorname{Log}}$ I have to solve $z^{1+i}=4$. Is there any easy way? I'm starting like this: $$e^{(1+i)\Log z}=e^{2\Log2}$$ Then I solve $$(1+i)\Log z=2\Log2$$ But I really doubt I ...
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3answers
63 views

Find all $z \in \mathbb{C}$ that satisfy $z^3 = −2(1 + i\sqrt{3})\bar{z}$

Find all $z \in \mathbb{C}$ that satisfy $z^3 = −2(1 + i\sqrt{3})\bar{z}$. You must express your answers in the standard form. So far, I'm thinking of writing $z = a + bi$, but then I have to ...
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3answers
49 views

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ in my lecture notes somehow my lecture got from$-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ to ...
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2answers
73 views

Determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity

I need to determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity. At first, I expressed this number as ${\dfrac{3}{5}+\dfrac{4}{5}i}$. Then I tried to use a formula for $\sin{nx}$, where x = ...
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1answer
31 views

equation draws a circle in C

It be $a\in\Bbb C , b ∈\Bbb R$ and $|a|^2 > b$. Also, $a'$ is the conjugation of $a: a' = x - iy$ when $a = x + iy$ (and equally for $z$). It needs to be shown, that the solutions of the equation: ...
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0answers
29 views

What's the name of the form (123i + 321)

Okay, so $0.5$ can be written as a fraction $\frac {1}{2}$. Is there an official name for writing a number in the form of $ai + b$? Complex numbers could be written in this form $z = a\ e^{i ...
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1answer
11 views

what will be the PDF of the magnitude of this random variable x+j y?

if we have a complex random variable [x+j*y] where (j :sqrt(-1)) and x,y both have Gaussian distribution and statistically dependent , so what will be the distribution (PDF) of the magnitude of this ...
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1answer
45 views

Path integral in the complex plane

Evaluate $\int_Tz\,\mathrm dz$ and $\int_T\overline z\,\mathrm dz$ where $T$ is the triangle with vertices $0,1,-i$ oriented clockwise. I am trying to solve this question, but I'm unsure how to ...
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2answers
44 views

Find 2 imaginary numbers that have a cosine of 4, using $\cos z =\frac{e^{iz}+e^{-iz}}{2}$

Use the definition $$ \cos z =\frac{e^{iz}+e^{-iz}}{2} $$ to find $2$ imaginary numbers having a cosine of $4$. I tried two approaches, both of which ended in failure: $$ 8=e^{iz}+e^{-iz}\\ ...
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0answers
21 views

Solve complex equation with exponentials

I have to solve: $e^z+2i=2e^{-z}$ I multipply both sides by $e^z$ and have: $(e^z)^2+2ie^z-2=0$ Now substitute $x=e^z$. $x^2+2ix-2=0$ $\Delta = -4+8=4$ so $x_1=-i-1$ and $x_2=-i+1$ Now go back ...
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2answers
52 views

Solving $(1-x)^3 = -1$ over the complex field

What are the solutions of: $(1-x)^3 = -1$ over $\mathbb{C}$? We have one real solution which is $2$ so there are two complex solutions.
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2answers
42 views

If $(1-i)^n = 2^n$ , then find $n$.

If $$(1-i)^n = 2^n$$ then find $n$. If anything raised to $0$ is $1$, but according to my book $ n \ne 0$. Is the print wrong?
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0answers
33 views

Quadratic formula and complex numbers

Let $az^2+bz+c=0$ be a complex quadratic equation. We know that it has $2$ roots: $z_1=\frac{-b-\sqrt{b^2-4ac}}{2a}$ and $z_2=\frac{-b+\sqrt{b^2-4ac}}{2a}$ If $b^2-4ac=1+i$ for example we have to ...
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0answers
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COMPLEX NUMBERS VERY HARD QUESTION…help me !!! [closed]

w and w^2 are complex cube roots of unity, a,b,c,d are real nombers If (a+w )^(-1) + (b+w )^(-1) + (c+w )^(-1) + (d+w)^(-1) = 2 w^2 ( a + w^2 )^(-1) + ( b + w^2 )^(-1) + ( c + w^2 )^(-1)+(d + w^2 ...
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1answer
37 views

synthetic division with $i$ in divisor

I divided $x^3-4x^2+4x-16$ by $-2i$ using synthetic division and got a remainder of $-8i-8$. Is that right? I'm not sure I'm doing this right.
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0answers
14 views

How to express vectors with more than 2 components in complex coordinates

It is straightforward to extend the notion of a 2D vector in the Cartesian x,y plane to 3D (x,y,z) or to any D. Sometimes it is useful to express vectors in the complex plane, where the 2D vector has ...
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1answer
21 views

Changing exponent sign

Sorry for the bad title, I am not sure how do I name it. Find all the roots that satisfy $z^4$ $$z^4 =\frac 12 e^{-i{\frac π7}} $$ $$z^4 = \frac 12 e^{i{\frac {13\pi}7}} $$ Therefore, the roots are. ...
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2answers
55 views

What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that: A holomorphic function is a complex-valued function of one or more complex variables that is complex ...
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1answer
13 views

$e^z=-3i$ find $z\in \mathbb C$ check my answer

I am unsure of my solution to this question, since the definition of the complex logarithm is somewhat complex. Since $-3i = 3e^{i\frac{3}{2}\pi}$ we get that $e^z=3e^{i\frac{3}{2}\pi}$ So if we use ...
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0answers
32 views

How to solve the complex equation? [closed]

$(a ^ b) ^ 2 = (m + in) ^ 2$ this will give $a \wedge b$ values. Can you please help me to solve this?
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1answer
36 views

Null Space of Transformation

I am given that $V$ is n-dimensional vector space over $\mathbb{C}$ and $T \in L(V)$. And $T$ has least $m$ distinct nonzero eigenvalues. How do I show that $\text{null}(T^{n-m}) = ...
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2answers
49 views

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number?

How to solve $|z^2-1|<|z|^2$ where $z$ is a complex number? I have tried it both with cartesian and polar coordinates but did not get a solution. I got that far: $z=x+yi$ and then I got: $$\pm x ...
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4answers
75 views

Ordering of the complex numbers

The complex numbers as a whole cannot be ordered but could you order the complex numbers of the form ai where a is a real number?
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0answers
51 views

What are the cases when the power result is a complex number? [duplicate]

Reference question - Make $a^b$ to have a complex answer Considering I have $a ^ b$ where both are real numbers and that the complex result is achieved in case when $$a<0 \wedge b = ...
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1answer
120 views

Find all z $\in \mathbb{C}$ that satisfy $z^3 = -2\left(1+i\sqrt{3}\right)\bar{z}$ [closed]

Find all z $\in \mathbb{C}$ that satisfy $z^3 = -2\left(1+i\sqrt{3}\right)\bar{z}$. You must express your answer in standard form. Hi, I tried letting $z = x+yi$ and $\bar{z}=x-yi $ but the steps ...
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2answers
62 views

Solve $e^{z-1}=z$ with $|z| \leq 1$

I'm looking for solutions to $$e^{z-1}=z$$ when $z \in \mathbb{C}$ with $|z| \leq 1$. The obvious solution is $z=1$, but I don't know how to show that there aren't any others. This question is ...
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0answers
56 views

Application of Rouché's theorem to $e^{z-1}=z$

I am reviewing my complex analysis and I got stuck with an exercise about Rouché's theorem. It states: for $0 \leq C \leq \frac{1}{e}$, show that $Ce^z=z$ has exactly one root in the closed unit disc. ...
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1answer
18 views

How can this equation be simplified this way? Transmission line: Zin

I thought of putting this on the Electrical Engineering Exchange but I thought since this seems more mathematical than related to engineering I thought I should place it here instead. Question: Why ...
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0answers
18 views

Holomorphic funtions

Let $U$ be an open connected subset of $\mathbb{C}^n$, and $O(U)$ the ring of holomorphic functions on $U$. Prove that $O(U)$ is an integral domain. I have done If $fg\equiv0$ in $U$, then $f$ ...
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0answers
14 views

Converting sum of complex exponential to sum of cosine

So I am trying to convert the equation $$\sum_{k=-2}^2 \alpha_k e^{i \frac{2 \pi}{T_0} kt}$$ Where $\alpha_0 = 1$, $\alpha_1 = 2 \angle30^\circ$, $\alpha_{-1} = 2 \angle{-30^\circ}$, $\alpha_2 = 1 ...
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2answers
38 views

Sum function operation: coefficient.

I have problem with the sum: $$ \sum_{k=0}^n \dbinom{n}{k}(\cos \alpha)^k(i\sin \alpha)^{n-k}\,\, $$ Apparantly, I have an imaginary unit therefore I need to distinguish even and odd powers of $i$ to ...
1
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2answers
56 views

How do I find the real and imaginary parts of $\dfrac{1}{z^2}$? [closed]

Find the real and imaginary parts of $\dfrac{1}{z^2}$ where $z = x + iy$