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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42
votes
12answers
6k views

Why is the complex plane shaped like it is?

It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someome ...
2
votes
2answers
1k views

Finding the least value for points in a loci

The question goes like this: On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing the complex numbers $w$ satisfying $arg(w-2)=\frac{...
5
votes
3answers
151 views
+100

Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$

My question is too simple. We know all that if we define the exponential function on $\mathbb{C}$ then we define the real part and imaginary part of $\exp{it}$ as $\cos{t}$ and $\sin{t}$. So if we ...
1
vote
1answer
17 views

locus of complex number 2

Que: If $arg(\frac{z-z_1}{z-z_2})=\pi$ then what is the locus of $z?$ Doubt In my textbook it is written that it represents the straight line joining $A(Z_1)$ and $B(Z_2)$ but excluding the ...
0
votes
0answers
39 views

Can we solve for $c$ in the equation $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0$?

Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. Suppose we have \begin{eqnarray} \sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\...
-2
votes
0answers
28 views

Given $y^2=4a(x-a)$, find expression for $\frac yx$ in terms of $y$ and show that $\frac yx \le 1$. [on hold]

Given $y^2=4a(x-a)$, find expression for $\frac yx$ in terms of $y$ and show that $\frac yx \le 1$. $a$ is a positive real number and $x\ge a$. This question is under the topic of complex numbers.
0
votes
2answers
50 views

If $ f $ has pole at $0$ then show that $e^f$ can't have pole at $0$.

i am trying to show that if $ f $ has a pole at $0$ then $ e^f $ can't have removable singularity at $0$ ? I tried to show that but i have a problem . I assume that $e^f$ has removable singularity ...
0
votes
0answers
24 views

Circle and line construction of a compex number $z\in\mathbb C$

Let $C\subseteq\mathbb C$ be the field of constructible complex numbers; that is, it includes only the elements $z\in\mathbb C$ which can be constructed with circles and lines. The field $E\subseteq \...
1
vote
2answers
44 views

Is this simple looking complex expression valid always?

$$ z^{a+ib} = z^a*z^{ib} \hspace{2mm} \forall z\in \mathbb{C} $$ In high school I was always taught to see the + in complex numbers as analogous to that is reals. However can it be proven to be ...
2
votes
1answer
147 views

How does one show that $\cos {\left (\ln 2 \right )}\approx \frac{10}{13}$?

How does one approximate the value of something like this? Apparently Euler found the value of $\large \frac{2^i+2^{-i}}{2}\large $ [which equals $\cos {\left (\ln 2 \right )}$] to be close to $\...
-1
votes
1answer
19 views

Deriving hyperbolic functions

If $tan(\theta+i\phi)$=$sin(\alpha+i\beta)$. Prove that $sin2\theta*cot\alpha$=$sinh2\phi*coth\beta$. i approached by taking $tan(\theta-i\phi)$=$sin(\alpha-i\beta)$ and then i found $tan(\theta+i\...
0
votes
2answers
39 views

How to prove main argument formula for any $z\in\mathbb C^*$

I would prove that for any complex number $z \in \mathbb C^*$ such that $z = x + \mathbb i y$ with $(x,y)\in\mathbb R^2$ and $x+\vert z\vert \neq 0$: $$ \arg z = 2\arctan\left(\dfrac{y}{x+\vert z\vert}...
0
votes
0answers
14 views

Prove that these pairs of complex numbers have real part 1/2 if they are symmetric in the complex plane.

Let matrix $A$ be defined as: $\Large A(n,k)=k^{-a_k + 1/2 + ib_k}$ if $k$ divides $n$, else $A(n,k)=0$ Let matrix $B$ be defined as: $\Large B(n,k)=\mu(n) n^{a_n+1/2 -ib_n}$ if $n$ divides $k$, ...
4
votes
1answer
63 views

Is it true that $ \sqrt{z^2-1} = i \sqrt{1-z^2} $?

I have seen a lot of times in books or on the internet that$ \sqrt{z^2-1} = i \sqrt{1-z^2} $ and I don't understand why that is correct . In general it is not true that $ \sqrt{-z}=i \sqrt {z}$ and ...
44
votes
8answers
94k views

How do I get the square root of a complex number?

If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
1
vote
2answers
40 views

Complex series should sum to zero but it's a puzzle

If we have a finite sum defined as $$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$ (where $k$ is an integer and $N$ is divisible by $4$), then how can we show that this sum is equal to $...
1
vote
2answers
26 views

Magnitude of complex number $a=\frac {1-e^{-i\omega L}}{1-e^{-i\omega}}$

I tried using $\sqrt{a*a^*}$ but I still got some complex parts...shouldn't the magnitude contain no complex part? I did: $a*a^*=\frac {1-e^{-i\omega L}}{1-e^{-i\omega}}\frac {1-e^{i\omega L}}{1-e^{i\...
1
vote
2answers
45 views

What does it mean to perform calculus upon functions of complex values? [on hold]

Complex numbers exist in a plane. This would lead me to believe that calculus views them as multivariate, but I am not real sure. How would one define a rate of change for a complex number valued ...
3
votes
0answers
92 views

Sine identity involving (3/p) for prime p greater than 3.

I am working through Ireland and Rosen's "Classical Introduction to Modern Number Theory" and am very stuck on this problem (#34 in Chp 5, 2nd edition): Note that $(a/b)$ is the Legendre symbol (or ...
1
vote
4answers
58 views

What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane?

What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? Is it $$\frac{-1}{4i}[z_1(z_2^* - z_3^*)-z_1^*(z_2-z_3)+{z_2(z_3^*)-z_3(z_2^*)}]$$ where $w^*$ denotes the ...
2
votes
4answers
91 views

A question about the definition of $\mathbb{C}$

In the usual definition of the field $\mathbb{C}$, as $\{(a,b):a,b\in\mathbb{R}\}$, the field $\mathbb{R}$ is not exactly a subset of $\mathbb{C}$, but only an isomorphic copy of the subfield $\{(a,0):...
28
votes
7answers
7k views

Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?

I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, ...
0
votes
2answers
42 views

Argument of complex numbers

If $z=re^{i\theta}$ and $w=\rho e^{i \phi} $ are two complex numbers, then $ arg(zw)=arg (z)+arg (w)$ But if $z=-1$ and $w=-1$, we get $ 0= 2\pi $ which is not correct. So why it gives us this ...
7
votes
1answer
355 views

Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
2
votes
1answer
34 views

Logarithmic function in complex number [on hold]

Show that: $$\cos[i\log(2+\sqrt3)]=2$$ I attempted by taking$(2+\sqrt3)$ into trigonometrical form but i am stuck Please help me out.
0
votes
3answers
54 views

A geometric approach to this problem?

Question: A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane ...
4
votes
3answers
126 views

What is the geometric interpretation of $|z-1|^2+|z+1|^2=4$ for all $z$ such that $|z|=1$?

Show that $|z-1|^2+|z+1|^2=4$ for all z such that $|z|=1$. [Note that $|z|$ refers to the magnitude of z where $z=a+bi$]. I was able to 'prove' the question; however, I cannot think of a geometric ...
0
votes
1answer
22 views

What is a geometric interpretation of multiplication/division in the complex plane? [duplicate]

How can one visualize the multiplication/division of a complex number, z, by a real number, an imaginary number, or another complex number?
3
votes
1answer
32 views

What is the kernel of $\phi$?

Let $\phi: \mathbb{C}^* \to \mathbb{R}^*$ with $z \mapsto |z|$ be a homomorphism. What is the kernel of this homomorphism? We know the identity in $\mathbb{R}^*$ is $1$. So we need to find the ...
1
vote
1answer
34 views

Show the limit exists.

For $|z|\neq 1$,show that the following limit exists: $$f(z)=\lim_{n\to\infty}\frac{(z^n -1)}{(z^n+1)}$$ Is it possible to define f(z) when $|z|\neq 1$ in such a way as to make $f$ continuous? ...
1
vote
1answer
35 views

Write $\,-4i\,$ in polar form

Write $\,-4i\,$ in polar form ${re}^{i\theta}$, with $r$, $\theta\in \mathbb R$, and $\,r\geq0,\;0\leq\theta<2\pi$. I let $\,z=-4i\,$ first, then get $\,r=\sqrt{0+{4^2}}=4$. However, $\,\tan\theta\...
2
votes
4answers
172 views

“Exponential Madness” (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
1
vote
3answers
175 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
0
votes
0answers
10 views

Modulus of complex number arrangement

Hithere, I have solved my equation to a point where I have: $$z_1 z_2 = 8r cis\frac{5\pi}{4}$$ $$\vert z_1 z_2 \vert = 2$$ Would I be correct in saying the modulus is 8r, thus $8r = 2$, so $r= \...
0
votes
1answer
25 views

Maximal value of real part of holomorphic function

Let $f:U \rightarrow C$ be a non-constant holomorphic function. $U$ is open, connected and $D(0,1+\epsilon) \subset U$. I'd like to show that there exists $z_0 \in \partial D(0,1)$ such that $Re(f(z))...
1
vote
0answers
65 views

Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ...
385
votes
20answers
66k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
2
votes
0answers
29 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
3
votes
2answers
52 views

Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right) $$ where I taken ...
1
vote
3answers
2k views

For complex $z$, find the roots $z^2 - 3z + (3 - i) = 0$

Find the roots of: $z^2 - 3z + (3 - i) = 0$ $(x + iy)^2 - 3(x + iy) + (3 - i) = 0$ $(x^2 - y^2 - 3x + 3) + i(2xy -3y - 1) = 0$ So, both the real and imaginary parts should = 0. This is where ...
1
vote
1answer
80 views

Is it true that $ \sqrt{-z} = i \sqrt z $?

Is it correct to write $ \sqrt{-z} = i \sqrt z $ , for every complex $z$? I think it's not true but I have seen it in some books . The reason I think it's not correct is for example if $z=i$ then $\...
1
vote
2answers
55 views

Calculating the gcd of complex numbers

I need help in calculating the gcd of complex numbers For Example: $\gcd(3+i,1-i)$. The problem is,I don't even know what's the algorithm for complex numbers...
0
votes
2answers
74 views

High powers of complex numbers [closed]

I have these two questions that I am trying to solve. I know that I am suppose to use De Movire's Theorem but I am getting stuck. Can you guys please help out? Thanks. Compute the following ...
5
votes
4answers
280 views

Easy partial fraction decomposition with complex numbers

There is an easy method to perform a partial fraction decomposition - described here, under the "Repeated Real Roots" title, for the coefficient A2. The problem is ...
1
vote
1answer
20 views

Inequality of absolute value of a complex number

If $ z $ is a complex number, does it follow that $ |z| \ge z $ like with real numbers? The way I justify it is by saying that if $ z \in \mathbb{C}, $ then $ z = a +bi $ for some $ a,b \in \mathbb{R}....
0
votes
3answers
31 views

Triangle inequality for complex numbers

I just start to learn about complex numbers and I want to prove the triangle inequality, which says that if $ z $ and $ w $ are complex numbers, then $ \displaystyle |z + w| \le |z| + |w|. $ My ...
0
votes
1answer
35 views

Need help finding conjugate

$$\overline{z-2+4i} = 2z+3+8i$$ I got this question on my online assignment. I got to a point where I couldn't get rid of the conjugate of z and I don't know how to expand or what to do with it. I ...
2
votes
2answers
47 views

How to sketch the region on the complex plane? [duplicate]

I am going through a basic course on complex analysis. I have a problem in understanding the following. E $\subset\mathbb{C}$ is defined as $$E := \{z\in\mathbb{C}:\vert z+i \vert = 2\vert z\vert \}$$ ...
1
vote
1answer
36 views

graphing a circle in the complex plane? [closed]

The ellipse seemed rather simple: Defining the equation of an ellipse in the complex plane But Wolfram won't graph it with equal axes. http://www.wolframalpha.com/input/?i=abs{%28x%2Biy%29}%2Babs{%...
1
vote
2answers
50 views

Automorphism of unit disk without zero

Let $S$ be the unit disk without $0$. Find all $f \in Auto(S)$ I got the following idea. By Riemann 0 is a removable singularity. Since for $g\in Auto(D)$ where $D$ is the unit disk. $g(z)= e^{i{\...