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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6
votes
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
16 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
17 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
48 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
17 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
37 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
vote
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
76 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
vote
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
votes
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
votes
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 = ...
-2
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
99 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... $\log(i+1)\log(i)=[\log \sqrt2+i(2n\pi+\frac{\pi}{4})]*i(2n\pi+\frac{\pi}{2})$
0
votes
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
132 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
65 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
24 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
52 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
37 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: ...
1
vote
0answers
26 views

To find minimum and maximum value of $|z|$

If $z$ be a complex number such that $\left|z+\frac{1}{z}\right|=a$,then prove that $\frac{\sqrt{a^2+4}-a}{2}\leq |z|\leq\frac{\sqrt{a^2+4}+a}{2}$. My attempt:$\left|z+\frac{1}{z}\right|\leq ...
1
vote
0answers
37 views

How can I plot the complex function in 2D?

My function: $$sin(wt-jT) \tag{1}$$ where $j$ - complex unit, $T=0.1,\ w=8 \pi,\ t=[0,0.01,0.02..100]$ I transform it to function with real arguments: $$\sin(wt)\cosh(T)+j\cos(wt)\sinh(T) ...
3
votes
2answers
117 views

Does $\overline{ \sqrt{1 + i}} = \sqrt{1-i} \ $?

I am having trouble with complex conjugates today. Can someone help me? $$\overline{ \sqrt{1 + i}} \stackrel{\color{#2222FF}{?}}{=} \sqrt{1-i} \tag{$\ast$} $$ In this case, since $\cos ...
2
votes
1answer
57 views

Solve $z^4+2z^3+3z^2+2z+1 =0$

Solve $z^4+2z^3+3z^2+2z+1 =0$ with $z$: a complex variable. Attempt at solving the problem: We divide the polynom by $z^2$ and we get: $z^2+2z+3+\dfrac{2}{z}+ \dfrac{1}{z^2}=0 $ $ $ We ...
2
votes
2answers
39 views

Is $z \mapsto \operatorname{Re}(z)$ a linear map?

I am referring here to the function that maps a complex number $z$ to its real part. This may be an obvious question, but it seems to me that it is; however, I wouldn't really know how to go about ...
1
vote
2answers
65 views

Prove that the exponential $\exp z$ is not zero for any $z \in \Bbb C$

How can the following been proved? $$ \exp(z)\neq0, z\in\mathbb{C} $$ I tried it a few times, but i failed. Probably it is extremly simple. If a draw the unit circle and then a complex number ...
7
votes
8answers
768 views

Obtain magnitude of square-rooted complex number

I would like to obtain the magnitude of a complex number of this form: $$z = \frac{1}{\sqrt{\alpha + i \beta}}$$ By a simple test on WolframAlpha it should be $$\left| z \right| = ...
2
votes
3answers
64 views

Prove that $\sum^{\infty}_{n=1}\frac{1}{n}\left(\frac{1+i}{\sqrt2}\right)^n$ converges but does not absolutely converge.

Prove that $\sum^{\infty}_{n=1}\frac{1}{n}\left(\frac{1+i}{\sqrt2}\right)^n$ converges but does not absolutely converge. My approach so far was to notice that ...
1
vote
2answers
21 views

Complex roots in order to apply residue theorem

$$\int_{0}^{2\pi}\frac{d\theta}{(4 + 2\sin\theta)^2}$$ $$\sin\theta = \frac{z - z^{-1}}{2i}$$ $$d\theta = \frac{dz}{iz}$$ $$\oint_c\frac{dz}{iz\left(4 + \frac{z - z^{-1}}{i}\right) ^2}$$ ending up ...
0
votes
0answers
19 views

Cauchy-Riemann: am I applying the equations correctly?

$$\text{Re}(z) +i\text{ Im}(z)^2$$ The problem states to apply C-R and to describe what can be concluded. However, I don't understand what I can conclude without a point $z_0$. My conclusion: ...
0
votes
0answers
26 views

Extension of complex numbers to higher dimensions without losing commutativity and multiplicative inverse

We have got the natural numbers $\mathbb{N}$ and by trying to achieve algebraic closure, the real numbers $\mathbb{R}$ and finally the complex numbers $\mathbb{C}$ were developed. I know the ...
1
vote
1answer
20 views

Applying the residue theorem on a real integral

$$\int\frac{d\theta}{a + b\cos\theta}$$ Given that $$\cos\theta = \frac{z + z^{-1}}{2}$$ $$d\theta = \frac{dz}{iz}$$ We have $$\oint_c \frac{dz}{iz\left(a + b\frac{z +z^{-1}}{2}\right)}$$ ...
2
votes
2answers
91 views

Solve $z^6=(z-1)^6$.

In the answer of Surb here : How to solve for the complex number $z$? I don't understand the subtlety. To me it's natural to do $$z^6=(z-1)^6\iff \left(\frac{z}{z-1}\right)^6\iff ...
0
votes
0answers
37 views

Integral over unit disk

I want to solve following integral $$\int_{|z|<1}\,\,\frac{1}{\sin\frac{1}{z}}\,dz$$ I know for every $\displaystyle z=\frac{1}{n\pi}$ ($n>0$) there exist a zero $\displaystyle z=\frac{1}{n\pi}$ ...
2
votes
1answer
30 views

Finding a simple fractional expression of a bilinear transformation given two fixed points.

If $z_1$ and $z_2$ are distinct fixed points of a bilinear transformation $w=T(z)$ show that the transformation may be expressed as $$\frac{w-z_1}{w-z_2}=K\frac{z-z_1}{z-z_2},$$ where $K$ is a ...
1
vote
2answers
77 views

How to solve $\sqrt {35 - 5i}$ [duplicate]

Need some hints on how to Solve $\sqrt {35 - 5i}$ Attempt. I factorized 5 out and it became $\sqrt {5(7-i)}$ I just want to know if it can be solved further. Thanks.
-4
votes
3answers
80 views

How to solve for the complex number $z$? [closed]

How to solve for the complex number $z$? $$z^6=(z-1)^6$$
1
vote
1answer
43 views

Limits and L'Hopital

$$\lim_{z \to i} \frac{z^4-1}{z-i}$$ I'm reading in a bunch of places that I can't use L'Hopital's rule for this problem. Why is this so? And if I can't use this rule then how would I go about ...