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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0
votes
1answer
35 views

Application of Rouche's theorem on $x^4-6x+3$

I'm being asked to find the number of zeros given $|z|<1$ and $1<|z|<2$. So here are my inequalities but I'm not quite sure how to find the number of zeros though based on these inequalities. ...
3
votes
1answer
147 views

Prove that the integral of $x\cos(x)/(x-2)(x-1)$ from negative to positive infinity is $\pi(\sin1-2\sin2)$. Use an indented contour

To do this I used the Residue Thm but the main issue here is that I cannot get the sine term to appear. Perhaps I'm ignoring something here. We know that the singularity is $x=1,2$ so we should just ...
0
votes
1answer
40 views

Expand $(e^{2x}-1-2x)/x^5$ into Laurent Series on 0<|x|<$\infty$ and classify its singularity

I guess I'm having difficulty with this because its not in the form of a polynomial expression, which is what I've been taught. Nevertheless here's what I did: I know that the expansion for ...
0
votes
1answer
79 views

Complex number, power series

Develop $\sinh z$ in powers of $z-\pi i$ to show that $$\lim_{z\to \pi i}\frac{\sinh z}{z-\pi i}=-1$$ I know that $\sinh z=\sum_{n=1}^\infty \frac{z^{2n-1}}{(2n-1)!}$. Edit: Following the hint ...
1
vote
3answers
86 views

Prove that the integral of $\sin^2(x)/(5+3\cos(x))$ from $0$ to $2\pi$ is $2\pi/9$

I'm not really unsure of how to approach this problem. I was thinking of reparametrizing the sin and the cos to its exponential form but I realize that it becomes even messier and leads sort of ...
0
votes
5answers
119 views

Complex numbers equivalence proof

I stumbled upon this exercise and can't seem to have any fruitful attemps, or rather I can't put together missing links. Let $z_1, z_2, z_3$ be different complex number such that $|z_1|=|z_2|=|z_3|$. ...
-1
votes
1answer
32 views

Evaluate the integral of $e^{x}/(x+1)^4$ on $\rho$, which denotes the entire imaginary axis

I'm not entirely sure if my intuition is correct but the singularity for this equation is -1 but -1 does not exist on the imaginary axis, so does this integral equal 0? If not, what am I missing and ...
-1
votes
1answer
70 views

Prove that the integral of $sin(z)/(z^2+4z+5)$ from negative to positive infinity is $-\pi sin(2)/e$

I think I've made the problem a lot nastier than it supposed to look. Here's what I have so far. First notice that $(z^2+4z+5)$ is equivalent to $(z^2+4z+4)+1$ so our singularities are -2-i and ...
0
votes
2answers
77 views

Prove that $\int_0^{\infty} \frac{x^2}{x^4+5x^2+4}dx = \frac{\pi}{6}$

Prove that $\int_0^{\infty} \frac{x^2}{x^4+5x^2+4}dx = \frac{\pi}{6}$ Obviously you would use Residue Theorem to tackle this problem. The correct answer to this is $\frac{\pi}{6}$ however I'm ...
1
vote
1answer
35 views

Minimum value of $\lvert z_1-z_2\rvert $ given $\lvert z_1-i\rvert ^2=4$ and $\lvert z_2-6\rvert =\lvert z_2\rvert $

Minimum value of $\lvert z_1-z_2\rvert $ given $\lvert z_1-i\rvert ^2=4$ and $\lvert z_2-6\rvert =\lvert z_2\rvert $ The answer is supposed to be $1$, but I keep getting $0$ when I graph the problem. ...
6
votes
4answers
442 views

Imaginary fraction square root?

I have a fraction - $-\frac{1}{3}$ Which could either mean the value of fraction is $\frac{-1}{3}$ or $\frac{1}{-3}$ Note the minus sign Now, what is the sqaure ...
1
vote
2answers
95 views

How to calculate $\zeta(i)$?

As the title says, I'm interested to know how $\zeta(i)$ is calculated. I know the functional equation for the zeta function, but if I put it in that in there, I must know $\zeta(1-i)$. Is it a good ...
0
votes
0answers
22 views

I always have some doubts regarding the inequalities in cases where the function become Complex in the field for the real numbers

Consider this inequality $x + \log\left(x \right)> \log\left (x\right) - 2$ Does this inequality has $-1$ as its solution ? It will be very helpful for me.
0
votes
3answers
93 views

How to simplify if $a > 0$ and $\cos(a) < 0$ [closed]

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$
1
vote
4answers
109 views

Simplify your answer completely: $5i(1 + i)^2$

Simplify your answer completely: $5i(1 + i)^2$ I know the answer is $-10$, but I don't know how to get it. Things I tried: foiling $(1+i)^2$ and then distributing $5i$. distributing $5i$ into ...
1
vote
4answers
46 views

A seemingly simple property of complex numbers that won't submit.

Let $a,b\in\mathbb{Z}[i]$ such that $|b-a|>|b|$ and $|a|>|b|$. I want to show that the absolute value of the real part of $\frac{a}{b}$ is greater than $\frac{1}{2}$. For example, let ...
4
votes
1answer
176 views

Determinant of skew-hermitian matrix

Given a skew-hermitian matrix $A \in \mathbb{C}^{N\times N}$, then $A = -A^H = -(A^*)^{T}.$ We can also say that $A^T = (-(A^*)^T)^T = -A^*.$ Thus, when computing the determinant we get $$ \det(A) = ...
1
vote
3answers
130 views

Cubic root of unity

Is there anyway to solve this without substituting with the values? Prove that: $$\frac{1+10w^2}{1-2w} + \frac{2+17w}{2+3w} = 6$$. (Where $w$ & $w^2$ are the cubic roots of unity)
0
votes
1answer
60 views

Euclidean circle in complex plane

I am reading Anderson's Hyperbolic Geometry and am having trouble with one of the Exercises in Chapter 1: Consider the unit circle $\mathbb{S}^1=\{z \in \mathbb{C} \text{ s.t. }|z|=1\}$. Let $A$ be a ...
2
votes
3answers
36 views

Nonlinear system with complex numbers

Solve the following system under the complex numbers (without eulerian form or polar form) $$z^3 + w^5 = 0 \\ z^2 \bar w^4 = 1$$ I have found that $(\pm 1, \mp 1)$ satisfy the equations as well as ...
0
votes
1answer
50 views

Algorithm for finding Complex Eigenvectors?

I'm wondering if there's a fairly easy algorithm by which one can, by hand, find eigenvectors corresponding to complex eigenvalues for small matrices. Of course, one can always row reduce, but it can ...
1
vote
1answer
65 views

Complex analysis clarification

Let us look upon a complex number "z". $$ z=x+iy $$ The following are true: $$ \bar z = x-iy, \quad |z|=\sqrt{x^2+y^2} $$ Since $z$ is a complex number it is represented in the complex plane (I hope ...
0
votes
1answer
96 views

How to prove that $((ia-1)/(ia+1))^{ib} = e^{-2b\operatorname{arccot}(a)}$?

$$\left(\dfrac{(ia-1)}{(ia+1)}\right)^{ib} = e^{-2b \cot^{-1} a}$$ I am trying to prove this. I tried by rewriting the lhs by multiplying by the conjugate, but I keep messing up something. Any help ...
4
votes
2answers
63 views

What does it mean if the standard Hermitian form of complex two vectors is purely imaginary?

If $v,w \in \mathbb{C}^n$, what does it mean geometrically for $\langle v , w \rangle$ to be purely imaginary?
0
votes
2answers
65 views

Let $\omega=\cos \theta + i \sin \theta$. Find, in terms of $\theta$, the argument of $(1-\omega ^2)^*$

Let $\omega=\cos \theta + i \sin \theta$. Find, in terms of $\theta$, the argument of $(1-\omega ^2)^*$ I started by using De Moivre's theorem and making the conjugate. Let $\alpha$ be required ...
0
votes
0answers
39 views

Why is this function not holomorphic on a disk?

I have the complex-valued functions $$f_1\left(z\right)=Erf\left(R-z\right)$$ and $$f_2\left(z\right)=Erf\left(R-\lvert z\rvert \right)$$ Now, I'm told $f_1$ is holomorphic over the disk of radius ...
1
vote
1answer
56 views

Can't find solution for equation

$100z=a(z+i)(z-i)^2(z-2)+b(z-i)^2(z-2)+c(z+i)^2(z-i)(z-2)+d(z+i)^2(z-2)+e(z+i)^2(z-i)^2$ I already got $b=5+10i, d=-5+10i $ and $e =8$ by eliminating the factors using $z=i, z=-i, z=-2$ but I cant ...
1
vote
1answer
44 views

Laurent series, function representation

Write the Laurent series for the function $f(z)=\frac{1}{1+z}$ $1<|z|<\infty$ I did $$\frac{1}{1-z}=\sum_{i=0}^\infty z^n\rightarrow \frac{1}{1+z}=\sum_{i=0}^\infty (-1)^nz^n$$ Is it right? ...
2
votes
1answer
32 views

Trouble with an easy complex equation

For the following equation: $\frac{Aj}{100\sqrt{2}} -\frac{A}{100\sqrt{2}} +\frac{x}{200}-\frac{xj}{200} =0$ where $A$ is a constant and $j$ is the imaginary unit. I thought the solution would ...
2
votes
3answers
230 views

Factor $z^4 +1$ into linear factors

$z$ is a complex number, how do I factor $z^4 +1$ into linear factors? Do I write z in terms of $x+yi$ so that $z^4+1=(x+yi)^4+1?$
2
votes
1answer
77 views

Laurent series , function representation

Write the Laurent series around zero for the entire function $f(z)=z^2e^{3z}$ I'm a little confused on how to represent the complex functions by series, as I did in the calculation of real functions, ...
8
votes
5answers
937 views

If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = ...
1
vote
4answers
70 views

How is it possible to find *all* of the roots of a complex number?

I was asked to find every fourth root of the complex number $i$. Setting $z=i$, we get $z=e^{i\frac{\pi}{2}}=e^{i(\frac{\pi}{2}+2\pi m)}$, where $m$ is any integer. ...
2
votes
3answers
60 views

Find all complex numbers that satisfies this equation

Find all complex numbers that satisfies this equation $(z - 6 + i)^3 = -27.$ I found one of them being $ z = 3 - i $
1
vote
1answer
14 views

Argument of the function $2 \frac{\sin(x)}{x} + \frac{\sin(x/2)}{x/2} e^{-i x/2} $

Plot the argument (phase ) of the Complex function $$2 \frac{\sin(x)}{x} + \frac{\sin(x/2)}{x/2} e^{-i x/2}$$ This can be written as $$\frac{1}{ix} (e^{ix} + 1 - 2 e^{-ix})$$ Wolfram shows ...
0
votes
4answers
93 views

Find the roots of the polynomial? (Cardano's Method)

$y^3-\frac7{12}y-\frac7{216}$ This is part of Cardano's method, so I've gotten my first root to be: $y_1=\sqrt[3]{\frac7{432}+i\sqrt{\frac{49}{6912}}}+\sqrt[3]{\frac7{432}-i\sqrt{\frac{49}{6912}}}$ ...
-1
votes
3answers
76 views

Sum of series $\sin \theta+\sin 2 \theta+\sin 3\theta+\dots$ [duplicate]

I need to prove sum of series: $$\sin \theta+\sin 2 \theta+\sin 3\theta+\dots=\sum_{n=1}^\infty \sin n\theta$$ by using in the first place the complex numbers.
0
votes
3answers
96 views

Proof of an interesting property of complex conjugates.

In a mathematical methods class we were told the following: $z= \frac{2+i}{(1+i)^i}$ Then the conjugate of the expression on the left of the equation is the RHS with the signs of the i's reversed. ...
0
votes
3answers
54 views

Solving an equation algebraicly

Just wondering, how can ALL the answers be found to the equation: $2^\pi+\pi=2^n + n$ Obviously $\pi$ is a real solution, but how can I get this result; and can I obtain other solutions?
0
votes
0answers
36 views

prove $\{z_n\}$ converges where $z_n=f(z_{n-1})$

Let $f : \mathbb C \rightarrow \mathbb C$, $f(z) = \frac{1}{2}z^2 + 1$, and $c = 1 + i$. Let $\{z_n\}$ be the sequence defined by iterating $f$ on some initial value $z_0 ∈ C$ (that is, $z_n = ...
0
votes
1answer
33 views

How to tell whether a complex sequence converges?

How do you tell whether a sequence with complex parts converges? For example, what would you do to prove whether the sequence $z_{n}=\frac{n}{(1+i)^{n}}$ converges?
0
votes
1answer
20 views

Simplifying an function with complex numbers

Given that variables $h_i, h_d $ are $\in \mathbb{C}$ i.e complex numbers. Can the following be further simplified $$h_dh_d,+h_ih_i+h_dh_d^*+h_ih_i^*\leq 2h_ih_d+h_ih_d^*+h_dh_i^*\tag 1$$ I know ...
18
votes
11answers
546 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? [duplicate]

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
2
votes
4answers
235 views

Complex number, series

Show that $$\frac{1}{z^2}=1+\sum_{n=1}^\infty (n+1)(z+1)^n$$ when $|z+1|<1$ I'm having problems to resolve this type of exercise since my book has virtually no exercises of this type, these ...
1
vote
1answer
43 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
12
votes
3answers
242 views

Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?

I've been brushing up on my complex analysis recently, and I've come across a problem that's stumped me: What are the real and imaginary parts of $$\sqrt{i+\sqrt{i+\sqrt{i+\sqrt{i+\cdots}}}} ?$$ I ...
1
vote
1answer
78 views

Complex Number inqualities

Although the inequalities are not defined on complex numbers. But does the inequality $x < 4 + 5i$ be said to possess any solutions ? Where $ i = \sqrt{-1}$.
0
votes
2answers
57 views

Can some one explain why the answer to part a describes a circle, or part of it?

Problem Statement: The transformation $T$ from the complex $z$-plane to the complex $w$-plane is given by $w=\frac{z+1}{z+i}, z\neq i$. a) Show that $T$ maps points on the half-line $\arg ...
0
votes
2answers
55 views

Have I Correctly Defined the Set of Nonzero Complex Numbers $\mathbb{C^*}$?

If the set of complex numbers $\mathbb{C} = \{a+bi\mid a,b \in \mathbb{R}\}$, then what would be the definition of the set of nonzero complex numbers? Am I right in defining such a set as ...
1
vote
1answer
43 views

Why is (-1)^(2/3) equal to -1/2+(i sqrt(3))/2

Can someone please explain to me how $(-1)^{\frac2 3}$ can be written as $\frac {-1}{2}+\frac{i \sqrt3} 2$ ? Do you use the corrolation $(-1)^c = e^{(i c \pi)}$, where ${c}$ is a constant?