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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0answers
62 views

Using $\arcsin(z) = -i\log(i(z + \sqrt{z^2 - 1}))$ to compute $\frac{d}{dz}\arcsin(z)$

I have to use $\sin^{-1}(z) = -i\log(i(z + \sqrt{z^2 - 1}))$ to compute the derivative of $\sin^{-1}(z)$, $\frac{d}{dz}\sin^{-1}(z)$. Here is my process: $$\sin^{-1}(z) = -i\log(i(z + \sqrt{z^2 ...
1
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2answers
61 views

is there an non-constant analytic complex function that receives only real values ? [duplicate]

I want to find something like $f(z) = |z|$ (complex function that get's only real values) the problem here is that $f(z)$ is not analytic.
2
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2answers
331 views

Find formula of sum $\sin (nx)$ [duplicate]

I wonder if there is a way to calculate the $$S_n=\sin x + \sin 2x + … + \sin nx$$ but using only derivatives ?
0
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4answers
67 views

showing the function |z| is analytic

So i need to shwo that f(z) = |z| is analytic. All i really have avavailable to me really are the cauchy riemann equations. So with that being the case i guess the assumption that the partials are ...
1
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3answers
145 views

Solving an equation with hyperbolic functions

I'm trying to prove that for a given $s,t\in\mathbb{R}$ there exists $w\in\mathbb{R}$ such that $\cosh(t)e^{i(s+w)}+\sinh(t)e^{i(s-w)}\in\mathbb{R}$. How to solve this?
0
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1answer
34 views

A multiplication for complex number.

I just found a formula on our lecture notes which is as follow: $(a + bi)(c + di) = (ac − bd) + (ad + bc)i$. but when I compute this, my result is: $(a + bi)(c + di) = ac + adi + bci + bdi$ where ...
0
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2answers
27 views

Use this parametrization to compute the following integral.

Let $$\Gamma$$ be the circumference centered at 1-i of radius 5 and transversed once in the counterclockwise direction. Parametrize the contour $$\Gamma$$. Use this parametrization to compute the ...
0
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1answer
61 views

Parametrize the contours of integration where Gamma is arc of the circle of radius…

Parametrize the contours of integration and write the integrals in terms of the parametrizations. Do not calculate them. $$\int\frac{\bar(z)}{z^3}dz$$ where $$\Gamma$$ is the arc of the circle of ...
0
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1answer
32 views

Locus of a point represented by a complex number.

Let $z$ represent a complex number. Then the curve, on which roots of the equation $(z-i)^n = \sqrt{21} + 2i$ lies on a circle with center at ? (where $i = \sqrt{-1}$) My attempt : $z-i$ ...
0
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1answer
29 views

Complex no maximization

If $$|z_1|=2,|z_2|=3,|z_3|=4$$ Find the maximum value of $|z_1-z_2|^2+|z_3-z_2|^2+|z_1-z_3|^2$ I tried using its geometrical interpretation but didn't get the answer.Try give an answer using ...
0
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1answer
33 views

Solving for $z$ in the complex equation $\sinh z = c$ for particular values of $c$.

What values $z$ satisfy $\sinh(z)=-i$, and $\sinh(z)=-1$?
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3answers
46 views

Complex Number Maths [closed]

I have started Complex Number question. I have tried solving but I cant find the answer.Kindly Assist. Given that $z=\cos θ +i\sin θ$ Prove $z+\frac{1}{z}=2\cos θ$
4
votes
1answer
123 views

Are there exists an analytic function satisfying the following condition

Let, $D=\{z\in \mathbb C:|z|<1\}$. Then there exists a non-constant analytic function$f$ on $D$ such that for all $n=2,3,4,...$ (a) $f\left(\frac{i}{n}\right)=0$. (b) ...
0
votes
1answer
79 views

Parametrize the contours of integration

I am having a difficult time figuring this problem out: Parametrize the contours of integration and write the integrals in terms of the parametrizations. $$\int_{\Gamma} (3\bar{z}^2+2z^3)\,dz$$ ...
0
votes
1answer
50 views

Verifying Euler's Formula from trigonometry

I know the proof for the Euler's formula by writing $e^{iz}$ as a Taylor series and arrange the brackets so that I get: $e^{iz}=cos(z) + isin(z)$. But I wonder if there is another way from going from ...
3
votes
1answer
263 views

Complex number times conjugate equals square of modulus (proof check)

My textbook asked me to prove that a complex number $r\operatorname{cis}(x)$, denoted by $z$, when multiplied by its conjugate is equal to its modulus squared. I realise that the second half of my ...
2
votes
1answer
54 views

Structure of solutions of $f(z)^n - c = 0$

Let $f(z)$ be a second degree polynomial, $n \in \mathbb{N}$, and a constant $c \in \mathbb{C} \setminus \{0\}$ with $|c| < 1$. We have the equation \begin{equation} f(z)^n - c = 0. \end{equation} ...
0
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3answers
43 views

Is this set equal to 3 points?

I tried to describe the following set: $$ \{ z \in \mathbb C \mid \left| z - {\sqrt{2}\over 2}\right|^2 \left| z + {\sqrt{2}\over 2}\right|^2 = {1\over 4}\}$$ Can you please tell me if my ...
2
votes
1answer
115 views

Is $\sqrt{x^2} = x$?

Does the $\sqrt{x^2}$ always $= x$. I am trying to prove that $i^2 = -1$, but to do that I need to know that the $\sqrt{(-1)^2} = -1$. If that is true then all real numbers are imaginary, because an ...
0
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1answer
132 views

Definite integral of absolute value complex function

Seems pretty straight forward but absolute values have always given me headaches $$\int_0^1 |1 -t + it|^2$$ Now usually I get roots and split up the intervals for when the function is greater or ...
0
votes
1answer
62 views

Schwarz theorem in complex analysis.

Is there a version of the Schwarz theorem $ \partial_x \partial_y = \partial_y \partial_x $ in the theory of complex functions of several variables and complex analysis ? It would be nice that you ...
1
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3answers
64 views

Find the value of $z^n+1/z^n$ if $n$ is arbitrary natural number

If $z$ is a complex number satisfying $$z + \frac{1}{z} = \sqrt{3}$$ then for arbitrary natural number $n$, determine the value of $$z^n + \frac{1}{z^n}$$ I have tried it with $n=2,3,4$ but it ...
3
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1answer
2k views

Solve $\sin(z) = 2$

There are a number of solutions to this problem online that use identities I have not been taught. Here is where I am in relation to my own coursework: $ \sin(z) = 2 $ $ \exp(iz) - \exp(-iz) = 4i $ ...
1
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1answer
74 views

Prove that when $z = p$ is a solution of $az^3+bz^2+cz+d=0$, $z=-p^*$ is also a solution

Given that $z = p$ is a solution of the equation: $$az^3+bz^2+cz+d=0$$ where $a$ and $c$ are real constants while $b$ and $d$ are purely imaginary constants. Show algebraically that $x = -p^*$ ...
1
vote
0answers
68 views

How to find roots of polynomial $a z \overline{z} + \overline{b}z + b \overline{z}+c$?

Let $a,c \in \mathbb R$ and $b \in \mathbb C$ with $|b|^2 - ac > 0$. I am stuck trying to find roots of the polynomial $a z \overline{z} + \overline{b}z + b \overline{z}+c$. I know the formula for ...
0
votes
1answer
40 views

Complex number raised to a complex number equality

If $a$ and $b$ are complex numbers, for which $z = x+iy$ does the formula $(z^a)^b = (z)^{ab}$ hold? My approach is the use the principal arguements $z^c = e^{cLog(z)}$ but this leaves me stuck, I've ...
0
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1answer
206 views

Collinearity in the complex plane with the unit circle

(a) Suppose p and q are points on the unit circle such that the line through p and q intersects the real axis. Prove that if z is the point where this line intersects the real axis, then ...
1
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1answer
136 views

Factoring A Matrix Polynomial

I'm working my way through this paper: Down with Determinants! In Section 2 (pretty much right off the bat) he gives his determinant-less proof that every finite-dimensional complex linear operator ...
1
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1answer
49 views

Solving equation with complex roots.

I have the following question. My problem lies in (c). Question a) Find the three roots of the equation $(w+5)(w+8)(w+9)=360$. b) Let $z_0=\sqrt{-2+6i}$, where $i^2={-1}$. Show that the solutions ...
2
votes
1answer
93 views

Principal argument of $-2i$

How to write the principal argument of $-2i$ ? I cannot just write $-\pi/2$, altough it is obvious, I have to justify it somehow. Can I say that $\lim\limits_{x\to 0}\arctan\left({\frac ...
0
votes
0answers
104 views

Complex numbers subgroup

Consider the complex numbers $C = \{a + bi| a, b ∈ R\}$ where $i =\sqrt{−1}$ satisfies $i^2 = −1$. It is a fact (but you need not prove) that the nonzero complex numbers $C^∗$ form a group under the ...
1
vote
1answer
59 views

What function does this series represent?

A midterm I proctored recently showed that $$ \cos(\sqrt{x}) = \sum_{k=0}^{\infty} \dfrac{(-1)^k x^k}{(2k)!}$$ The question asked what function this series represents. It may represent cosine, but ...
0
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2answers
72 views

Partial sums of trig functions identity

Using the fact that $$\sum\limits_{k=0}^{n}z^k=\frac{1-z^{n+1}}{1-z}$$ I want to find the partial sum for multiples of the trig funtions, ie. $1+\cos(\theta)+\cos(2\theta)+\cdots+\cos(n\theta)$ and ...
1
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3answers
81 views

Proving cot identity using Euler identities

I'm trying to prove that $\cot(2\theta)+\csc(2\theta)=\cot(\theta)$. I'm using that $$\sin(\theta)=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})\qquad \cos(\theta)=\frac{1}{2}(e^{i\theta}+e^{-i\theta})$$ So ...
0
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2answers
512 views

Effect of a simple pole vs complex conjugate poles

If $H(s)$ is a transfer function and it has just one pole in $s = p$, $p \in \mathbf{R}$, $$H(s) = \displaystyle \frac{H_0}{(s - p)}$$ the frequency response is $20 \log_{10} |H(j\omega)|$. With ...
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votes
1answer
50 views

No branch point on composite function [closed]

I need to know why $f(z)=\sqrt z$ and $g(z)=\sin(\sqrt z)$ have a branch point at $z=0$ but $h(z)=\frac{g(z)}{f(z)}=\frac{\sin(\sqrt z)}{\sqrt z}$ has NO brach point at $z=0$
3
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1answer
368 views

Intersection of a line through two points on a unit circle with real axis

Suppose we are given two points on unit circle which are represented as complex numbers $u$, $v$. We want to show that the intersection of the line through $u$ and $v$ and the real axis is ...
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1answer
41 views

what is the greatest value of $|Z|$ for given equation…

How to solve this question? Any ideas on how to start?
1
vote
2answers
27 views

Find the locus of $w$

$$ \text{Find the locus of $w$, where $z$ is restricted as indicated:} \\ w = z - \frac{1}{z} \\ \text{if } |z| = 2 $$ I have tried solving this by multiplying both sides by $z$, and then using the ...
0
votes
1answer
20 views

Claim: The complex conjugate of $ \omega_N $

My textbook says that $ \omega^{N-k}_N = \bar \omega^k _N $, where the bar denotes the complex conjugate. Why is this true? Sidenote: I believe $\omega^{N-k}_N=(e^{-2\pi i}/N)^{N-k} $.
2
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5answers
65 views

how does this expression cancel out

How does $$\frac{2t-1-i}{2t^2-2t+1}=\frac{1+i}{-1+(1+i)t}$$ I just can't see how this works... I typed the LHS in WFA and it gave the RHS but I don't know how anything can cancel.
0
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3answers
53 views

I have a problem with this complex numbers equation

The question is how many roots $z$ has. How can i approach this, such that i will be able to solve this kind of problems? I thought to use de moivre formula. How can i apply this here ? $z^2 = ...
0
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3answers
35 views

Root of real and complex polynomial

Let $z \in \mathbb{C}$ be a root of real polynomial $p(x)=\sum_{k=0}^{n} a_k x^k$ ,$a_k\in \mathbb{R} \forall k$. How to proove, that $\overline{z}$ is also a root of given polynomial? Is that true ...
0
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2answers
36 views

Solve this Complex Equality $z^2+|z|^2 = \sqrt{2} z |z| $

I am trying to solve this complex equation. $z^2+|z|^2 = \sqrt{2} z |z| $ Trying to expand it by rewriting it in the (a+ib) form gets me stuck. Also, polar form does not seem helpful. Any ideas?
0
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1answer
42 views

Prove the following based on the triangle inequality

So I've just proven $|z_1+z_2|\le |z_1|+|z_2|$ and then I proved if $a=z_1+z_2$ and $b=z_2$ then $||a|-|b||\le|a-b|$ And now I have to prove the following: You can see that the top half is the ...
5
votes
1answer
312 views

Show that SO(2) is isomorphic with the complex circle group

For my math study I have to prove that $SO(2)$ is isomorphic with the complex circle group. Some steps in this prove are a bit difficult to me, so I hope you could help me. With $SO(2)$ I mean the ...
0
votes
1answer
19 views

Simplifying a complex term

How can I simplify this term $\frac{7+5i}{i-1}$? I've tried multiplying both numerator and denominator with $-1-i$, but in results in a division by zero.
0
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1answer
37 views

Simplifying $i |x|$

Is there any way that one could condense the expression $$i |x|$$ where $i$ is the imaginary unitto get the $i$ inside of the absolute value? I have not been able to find any way to do so but I feel ...
7
votes
3answers
96 views

Set Theoretic Definition of Complex Numbers: How to Distinguish $\mathbb{C}$ from $\mathbb{R}^2$?

I have spent some time looking for a rigorous, set-theoretic definition of the complex numbers. I have read the book Elements of Set Theory by Herbert Enderton (1977) which does an excellent job of ...
1
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2answers
120 views

Compute $(-3+4i)^{1+i}$

Compute $(-3+4i)^{1+i}$ Here is what I got. $z=-3+4i$ so $|z|=5$ and $\operatorname{arg}(z)=\pi -\arctan(-4/3)$ $$ \begin{align} (-3+4i)^{1+i} & =e^{(1+i)(\log*-3+4i)} \\ & =e^{(1+i)(\log 5 ...