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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1
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1answer
18 views

Invariant under $x \rightarrow 1/x$?

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that $\phi(z)=\phi(z^{-1})$ So far, all I could come up with was a family of ...
0
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1answer
27 views

Complex Numbers in Factoring [closed]

Why does "$i$" only get involved in factoring a function when there is a ($+$) in the equation? EX: $x^2 + 9$.
1
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1answer
17 views

How to express an angle in terms of pi

I have the complex number $z = 5 + 6i$ in polar form $$z = \sqrt{61} (\cos \theta + i\sin \theta)$$ and $$\theta = \tan^{-1}\left(\frac{6}{5}\right) = 0.87605805059 \text{ rad}$$ But I need that ...
1
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0answers
28 views

Find maximum value or upper bound of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$ [duplicate]

If $|z_{1}|=2,|z_{2}|=3,|z_{3}|=4$,then find maximum value of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$. My attempt:I considered 3 circles having centre origin and radii as $2,3,4$. Then I ...
-1
votes
1answer
100 views

$z^n=(i+z)^n$, solve for $z$

I came across this question from an older textbook with no answers and I'm a bit stuck. Currently, I have done the following; let $z=r(\cos x+i \sin x)$ ($z \in \mathbb{C}$) therefore, we now have ...
0
votes
2answers
43 views

Continuity of a function with complex variables

How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$? $$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i ...
2
votes
1answer
38 views

Calculate complex eigenvector

Hi i have problem i hope that someone can make this for me more clear: So i have matrix $A = \begin{bmatrix} -2 & 1 \\ -2 & 0 \\ \end{bmatrix}$ I have to calculate eigenvector as matrix $P$ ...
2
votes
3answers
36 views

Compute the integrals using the residue theorem

Compute the following integrals: $I:=\int_{|z|=2}\frac{1}{(z-3)(z^{13}-1)}dz$ $J:=\int_{|z|=10}\frac{z^3}{z^4-1}dz$ I do not know where to begin. I know I am supposed to use the substitution ...
0
votes
1answer
24 views

Find the real and imaginary parts in the given expression:

$$(z+1)^2=u(r,\theta)+iv(r,\theta)$$ We are learning how to apply the polar form of the Cauchy-Riemann equations. I understand how to do this using the C-R equations the non-polar way, but I'm at a ...
2
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1answer
27 views

How do I express this $f(x,y)$ in terms of $f(z)$

$$f(x,y)=e^y\sin x+ie^y\cos x$$ The problem requires me to express in terms of $z$ only. My attempt: $$=e^y(\sin x+i\cos x)$$ $$=ie^y(\cos x-i\sin x)$$ If $e^{-i\theta}=(\cos\theta - i\sin ...
0
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0answers
12 views

Adding a constant after finding harmonic conjugate

The question asked me to find the harmonic conjugate of $u$ and then express $f(x,y)$ in terms of $z$. I found the conjugate. I also reasoned that it must be the expanded form of $z^2$. However, they ...
0
votes
2answers
20 views

Is this function harmonic?

$$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to figure out if this function is harmonic. I worked out all the algebra and my conclusion is no because the numerator of the final fraction is $2x^3+2xy^2$ ...
0
votes
2answers
15 views

Radius of convergence for complex power series

I am supposed to find the radius of convergence for the complex power series $$\sum_{n=0}^{\infty}(-1)^n2^nz^{2n+2}$$ I know that the radius of convergence is calculated by ...
0
votes
1answer
17 views

Help finding a second homogeneous polynomial of degree 5 that are also harmonic

Essentially I have to find 2 homogeneous polynomial of degree 5 that are also harmonic. Knowing z=(x+iy) is analytic I found my first polynomial to be f(z)=z^5 and that multiples of this would ...
14
votes
1answer
234 views

Proving that $e^\pi=e^{-\pi}$

I've been stuck with this for a while now. I have this chain of reasoning that would imply $e^{-\pi}=e^\pi$, obviously false, since $e^\pi$ and $e^{-\pi}$ are two real distinct numbers and so I must ...
2
votes
3answers
46 views

Image of a family of circles under $w = 1/z$

Given the family of circles $x^{2}+y^{2} = ax$, where $a \in \mathbb{R}$, I need to find the image under the transformation $w = 1/z$. I was given the hint to rewrite the equation first in terms of ...
0
votes
1answer
28 views

Simplify: $2w(z^2+1)=2z(w^2+1)$, $z,w \in \mathbb{C}$

I'm trying to simplify: $$2w(z^2+1)=2z(w^2+1), z,w \in \mathbb{C}$$ Simplifies to: $$2wz^2-2zw^2+2w-2z=0$$ But doesn't seem like it would form a nice quadratic function. Any tips? Would it be ...
2
votes
1answer
52 views

If $(z_{n}) \in \overline{ \mathbb{C}}$, $z_{n} \to \infty$ as $n \to \infty$, what happens to $|z_{n}|$, $Re(z_{n})$, $Im(z_{n})$, $Arg(z_{n})$?

Suppose the sequence $(z_{n}) \in \overline{\mathbb{C}}$ (where $\overline{\mathbb{C}}$ is the extended complex plane) converges to infinity as $n \to \infty$. I need to determine what this implies ...
6
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1answer
52 views

Classification of homomorphisms $\mathbb Q \to \mathbb C^\times$

Are there any textbooks which discuss/classify the injective group homomorphisms from $\mathbb Q$ (under addition) into $\mathbb C \setminus \{0\}$ (under multiplication)?
-3
votes
0answers
29 views

Multiply 3 - 2i by its conjugate. [closed]

Please explain me, what is conjugate and how it is multiplied. Multiply $3 - 2i$ by its conjugate
0
votes
2answers
19 views

If $a, b\in\mathbb{C}$ and $ae^{2it}+be^{-2it}$ is real, then $ae^{2it}+be^{-2it}=a'\cos(2t)+b'\sin(2t)$.

I'm asked to show that If $a, b\in\mathbb{C}$ and $ae^{2it}+be^{-2it}$ is real, then $ae^{2it}+be^{-2it}=a'\cos(2t)+b'\sin(2t)$ for some $a',b'\in\mathbb{R}$ My work so far is as follows. ...
1
vote
1answer
18 views

Is the complex form of the Fourier series of a real function supposed to be real?

The question said to plot the $2\pi$ periodic extension of $f(x)=e^{-x/3}$, and find the complex form of the Fourier series for $f$. My work: ...
1
vote
1answer
15 views

What to do when there is only one valid value to be used in the Cauchy-Riemann equations

I just did 2 problems where the $u$ part of the C-R equation was $0$. I'll give one as an example. I'm confused as to what conclusions I can correctly arrive at. $$f(z)=Im(z)$$ So I can say that ...
3
votes
2answers
50 views

Representation of roots of unity.

How to represent solutions of $\sqrt[26]{1}$ with solutions of $\sqrt[26]{-1}$? I know that $$w_{k}=\cos\left(\frac{0+2k\pi}{26}\right)+i\sin\left(\frac{0+2k\pi}{26}\right), \; \; ...
0
votes
1answer
19 views

Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable.

$$ f(z) = e^{−x}e^{−iy}$$ I used the Cauchy Riemann equations to determine that $x=iy-\ln(i)$, but I'm not sure what I'm supposed to conclude. Could I say that the function is differentiable wherever ...
1
vote
2answers
38 views

Is it a removable singularity?

In the function: $$ f(z)=2iz\frac{(1-z^{2})^{\frac{1}{2}}}{1-2z^{2}} \qquad \qquad (z \in \mathbb{Z}) \,\, , $$ There is a singularity at the point $z=\pm \sqrt{1/2}$. Is that a removable ...
1
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0answers
23 views

Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ then is $f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$?

I am a little stuck here, suppose we have some function $$f(z)=f(x+iy)=u(x,y)+iv(x,y)$$ then is $$f'(z)=\frac{\partial u}{\partial x}(x,y)+i\frac{\partial v}{\partial x}(x,y)$$ assuming $f$ is ...
4
votes
2answers
77 views

Are there any other solutions to this equation?

Consider the equation $1-t = tx^{1-2t}$ for some complex number $t$ and real $x$. Are there any other solutions to this equation besides $\Re(t) = \frac{1}{2}$ ? My attempt: The above equation can be ...
0
votes
2answers
42 views

Complex Analysis: Show that $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$ [closed]

How can I show that if $|a|<r<|b|$, then $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$, where $\gamma$ is the circle with center the origin, radius $r$, and positive orientation? ...
1
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2answers
55 views

Two complex numbers can be equal but why can't they are greater or lesser?

Yes we know that two complex numbers can be equal to one another , but why can't we say that a complex number is greater/lesser from another complex number ?
3
votes
1answer
35 views

Relation between $|z^x|$ and $|z|^x$

In the answers given to this question, the following relation is often used: $$\left| z^x \right| = \left| z \right|^x$$ with $z \in \mathbb{C}$, $z = \alpha + i \beta$. How to prove it? Can $x$ ...
0
votes
1answer
23 views

$z = 1/w$ transformation for parallel lines $y = x + b$

I am supposed to find the image of the family of parallel lines $ y = x + b $ under the transformation $w = \frac 1 z $. Attempt: Replace $x$ and $y$ with $\Re(z)$ and $\Im(z)$, respectively. ...
0
votes
0answers
17 views

How to reflect a complex point A through complex point B?

I'm doing a math project and I was wondering if anyone could tell me the formula for reflecting a complex point B through point A? I know that across the x axis its the complex conjugate, but I wanted ...
2
votes
1answer
39 views

Writing N-th roots of unity

I have a question regarding roots of unity. In general, we can write the n-th roots of unity as $$e^{2\cdot\pi\cdot i\cdot\frac{k}{n}}$$. However, if we do the following manipulation we get the ...
0
votes
1answer
31 views

Does $\lim_{x \to 1^-} \sum_{n=0}^\infty x^{n!} = \infty$?

Does $\lim_{r \to 1^-} \sum_{n=0}^\infty r^{n!} = \infty$? I am working on a complex analysis question that asks to show $\sum_{n=0}^\infty z^{n!}$ cannot be extended past the open unit disk. My ...
0
votes
1answer
42 views

In the Set of Extended Complex Numbers, 3/0 = infinity?

There's something I don't quite understand about the extended set of complex numbers. Usually, a number $\frac a 0 , a \in R$ is undefined. However, in the set of extended complex numbers, $\frac a ...
0
votes
1answer
25 views

Find the limit or explain why it doesn't exist

I'm having trouble starting this problem. I don't understand how I can make the limit apply when it's written with $x$ and $y$. $$\lim_{z \to (1-i)} (x+i(2x+y))$$ If I change $z$ to $(x+iy) \to ...
0
votes
0answers
25 views

Find the harmonic conjugate of the following

I just want to make sure my reasoning is correct. I followed another similar question from this site. Also, is there a method I can use after coming to the conclusion below to check to make sure my ...
0
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0answers
21 views

Applying Cauchy-Riemann to $f(z)$

$$\ln|z|+i\text{Arg}(z)$$ the problem states that I have to apply Cauchy Riemann to the problem and determine a conclusion. Below is how far I got, but I'm not sure how to take the derivative of ...
0
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2answers
25 views

General way to express holomorphic function in terms of z?

For the holomorphic, complex-valued function f, defined as $f(x + iy) = xy - x + y + i(-(1/2)x^2 + (1/2)y^2 - x - y + c)$ We can express this in terms of $z$ and $\bar z$ by substituting $x = ...
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votes
5answers
111 views

Solve the following equation: $z^4+z^3+z^2+z+1=0$ [closed]

How do I solve the following equation: $z^4+z^3+z^2+z+1=0$
0
votes
0answers
105 views

Mapping in the complex plane

I have the following two circles in the complex plane, $z = x + iy$, which bound a region, $R$. The equations for the circles and a sketch of the region is given as follows: $$ x^2 + (y-1)^2 = 1\\ x^2 ...
1
vote
3answers
59 views

Find the general values and principal values of $i^{log(i+1)}$?

Find the general values of $i^{\log(i+1)}$? I tried this way... $\log(i+1)\log(i)=[\log \sqrt2+i(2n\pi+\frac{\pi}{4})]*i(2n\pi+\frac{\pi}{2})$
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2answers
19 views

A line within the complex plane or not

Totally new here on the site so... I'm having a bit of trouble with determining on whether or not a set of complex numbers is a line. The following is an example: Consider the set of all complex ...
1
vote
4answers
133 views

Complex number ( prove ) [closed]

Let $$ {x-yi\over{x+yi}}=a+bi\;\;. $$ Prove $a^2+b^2=1$ I don't know how to start prove it, can anyone help me?
0
votes
0answers
68 views

Mapping a Region Using an Exponential Function

I have the following two circles in the complex plane, $z = x + iy$, which bound a region, $R$. The equations for the circles and a sketch of the region is given as follows: $$ x^2 + (y-1)^2 = 1\\ x^2 ...
1
vote
1answer
26 views

Dense on the unit circle

I am reading: "It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)" How is this possible? Can ...
1
vote
0answers
53 views

Does $f:\mathbb{R}^d\to \mathbb{C}$ implies $|f|<\infty$ almost everywhere?

I was reading notes on measure theory, and just want some clarifications. If $f:\mathbb{R}^d\to [0,+\infty]$, is it allowed to have $f(x)=+\infty$? What does it mean? Because when I learned the ...
0
votes
1answer
38 views

Lower bound for a linear combination of two related complex numbers

I have been trying this problem for sometime. If $a, \;b,\;\alpha$ are complex numbers such that $|a|\leq K|b|,$ and $|\alpha|\leq 1$ where $0\leq K\leq 1,$ then I want to express the lower bound ...
0
votes
1answer
29 views

Cauchy riemann equations - determine a conclusion

$$(x^2+y^2)+2ixy$$ I'm supposed to apply the Cauchy Riemann equations to the above and figure out what conclusions can be made. Below is what I've ended up with after applying the equations: ...