# Tagged Questions

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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### Can principal curvatures be complex numbers in a real world situation?

Can the equation for the principal curvatures, $k^2 - 2Hk + K = 0$ (where H is equal to the mean curvature and K is equal to the Gaussian curvature), ever have complex roots? In other words, where ...
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### Finding $f^{(m)}(i)$ where $f(z) = (1 + (z - i)^2)^{-1}$ without differentiating.

I have a question and I'm not to sure how to approach it, so any kind of help will be awesome. I was given this question in the practice final, however there are no solutions/hints to this question, ...
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### Proving that $\mathbb{C}$ has a natural definition

I would like to prove a theorem in complex analysis which states: Let $K$ be a commutative field. We suppose that $L$ is a sub-field of $K$ and that $K$ is thus a vector space of finite dimension ...
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### does $\sqrt[i]{x}$ have infinite solutions like the complex logarithm?

We know that $\log(x)= z+ 2πix$ where $z$ is a solution to $\log(x)$ and for any integer $x$. Can a similar derivation be made for $\sqrt[i]{x}$? Intuition says yes because of a derivation of Euler'...
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### Couldn't there be a better definition of imaginary numbers? [closed]

It is a common known fact that i is defined as a number that satisfies the quality of $i^2 = -1$ or in other words, $i = \sqrt-1$. The intuition of this is often said to be that multiplying by $i$ ...
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### Residues and poles, proof with poles

Proof that statements i)If $f_1$ and $f_2$ have residues $r_1$ and $r_2$, show that the residue of $f_1+f_2$ at $z_0$ is $r_1+r_2$. ii)If $f_1$ and $f_2$ have simple poles at $z_0$ show ...
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### Complex analysis, find the residue

Find the residue of $f(z)=\frac{1}{z^2\sin z}$ at $z_0=0$ What I tried Let $g(z)=1$ and $h(z)=z^2\sin z$, both are analytics but they have zeros of different orders then $f(z)$ don't have removable ...
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### Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$

Let $z_1,\dots,z_n\in\mathbb{C}$ such that $|z_p|\le1$ for every $p\in\{1,\dots,n\}$. Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$. I have firstly tried ...
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### Residues and poles proof

Let the degree of the polynomials $P(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ $a_n\neq0$ and $Q(z)=b_0+b_1z+b_2z^2+\cdots+b_mz^m$ $b_m\neq 0$ be such that $m\geq n+2$. Show that if all the zeros of $Q(z)$...
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Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$ ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi i\sum_{n=0}^\infty\frac{... 1answer 35 views ### When is$1+e^{-i\pi(a+b)}+e^{-i\pi(b+c)}+e^{-i\pi(a+c)}$non-zero,$a$,$b$and$c$being integers? I am trying to find the conditions on the integers$a$,$b$and$c$such that $$1+e^{-i\pi(a+b)}+e^{-i\pi(b+c)}+e^{-i\pi(a+c)}$$ is not equal to zero. I think that the conditions for which it is equal ... 3answers 72 views ### Complex number$\frac{z}{z+1}=2+3i$Given that$\frac{z}{z+1}=2+3i$, find the complex number$z$, giving your answer in the form of$x+yi$. Can someone give me some hints for solving this question? Thanks 1answer 91 views ### Help with the proof that the sum of all the roots of a complex number is zero If a complex number$z \neq 0$has n roots, then each root can be expressed as: $$z_j=(\sqrt[n]{r}) e^{ {i (\theta +2\pi j) }/{n} }$$ For$j=0,1,2,...,n-1$Thus, the summation of all the roots ... 1answer 37 views ### Complex integrate and residues Evaluate the integral of that$f(z)=\frac{z+1}{z^2-2z}$around the circle$|z|=3$oriented counterclockwise First I found that singularity points are$z=0,z=2\star\int_{|z|=3}\frac{z+1}{z^2-2z}=2\... 1answer 33 views ### Complex analysis, residues Find the residue at z=0 of f(z)=\dfrac{\sinh z}{z^4(1-z^2)}. I did \begin{align} \frac{\sinh z}{z^4(1-z^2)} & =\frac{1}{z^4}\left[\left(\sum_{n=0}^\infty \frac{z^{2n+1}}{(2n+1)!)}\right)\... 1answer 22 views ### Question about a simple rule of the complex logarithm According to the Wikipedia page on complex logarithms: Also, the identity \log(xy) = \log x + \log y can fail: the two sides can differ by an integer multiple of 2\pi i. Does the same hold ... 1answer 51 views ### Complex Numbers and their Conjugates How many complex numbers are there that are conjugates of their own cubes? Is there some simple way to find this? 2answers 384 views ### Mathematical Way To find how many complex numbers [closed] Suppose that a and b are integers and that |a + bi| \leq 5 , then how many complex numbers a + bi are there? Is there a mathematical way to do this? 2answers 81 views ### What does this circle look like? I have been given an integral to evaluate over a circle|z-i|=3$$I am trying to work out what this circle looks like. Is the radius of the circle 3 centred on the origin? 2answers 94 views ### The Imaginary number [duplicate] Sometimes one writes i^2=-1 to characterize the imaginary number and sometimes as the root of -1. So when I resolve the first equation, I get for the imaginary number two roots of -1 and thus ... 1answer 146 views ### e^{2\pi i x} = (e^{2\pi i})^x: What happens if x is rational? [duplicate] I'm a bit embarrassed that I've had difficulty on getting around this one:$$e^{2\pi i x}$$Solving it by itself, we can reduce it down to (e^{2\pi i})^x = 1^x such that e^{2\pi i x} = 1 for all ... 1answer 36 views ### Norm of a complex cross product Let c=(c_1,c_2,c_3) be a complex vector. How can we see that \|c\|^2=\|c\times \bar{c}\|? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ... 2answers 84 views ### Complex Number, Quaternions and Octonions [duplicate] There are complex \mathbb C, quaternions \mathbb H and octonions \mathbb O. Is there any higher dimensional generalization of them, such in the \mathbb R^{16}? Or why do we just study three ... 1answer 99 views ### Real Manifold … Complex Coordinates? I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 ... 1answer 28 views ### What would the multifunctional inverse of F(x)=|x| be? What would the multifunctional inverse of F(x)=|x| be, assuming x is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ... 3answers 72 views ### Complex numbers? [duplicate] There are plenty of questions out there asking what complex numbers mean and I never seem to get any of them. I have a few specific questions i want to ask about complex numbers. 1) what is the ... 1answer 38 views ### Proving relation for square root of complex number How do I represent \sqrt{1 + ja}? I'm trying to show that it's approximately equal to \pm(1 + \frac{ja}{2}) when a \leq 1. 2answers 30 views ### Ignoring the pole? I have the integral$$\int^{2 \pi}_0 \frac{z}{z+2} dz$$where$$|z|=3$$I parametrise the integral and get$$\int^{2\pi}_0 \frac{9 ie^{2i\theta}}{3e^{i\theta}+2}$$and this gives the required answer ... 1answer 35 views ### Evaluating a complex integral in punctured plane. I am trying to evaluate the complex integral$$\int \frac{z}{z+2} dz$$And$$|z|=3$$We can see there is a pole at z=-2. How do I go about solving this, what is the strategy? 1answer 238 views ### Arithmetic progression with complex common difference? Suppose we have the following sequence:$$\{0,i,2i,3i,4i,5i\}$$Can we call this sequence an arithmetic progression with first term 0 and common difference of i ? Clarification: Here, i is ... 2answers 41 views ### Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers? I've been wondering if there is any use to defining a set that is isomorphic to \mathbb{Q}^2 (in the same way that \mathbb{C} is isomorphic to \mathbb{R}^2). I immediately see a problem with e.... 0answers 34 views ### Find a generator for vectorial subspace S = {(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0} c+d-i(2ia)=0 c+d+2a=0 c=-d-2a (a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1) Is this solution correct? 3answers 134 views ### Why imaginary numbers axis is plotted perpendicular to the real numbers axis? Negative numbers axis is plotted to the opposite side of the positive real number axis that make sense but i do not understand why imaginary numbers are plotted perpendicular to the real numbers axis. 4answers 2k views ### Cauchy integral formula Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! \begin{... 1answer 211 views ### Midpoint of two complex numbers in polar form Say we have two complex numbers: re^{i\theta} and se^{i\phi} Is there a straightforward way to find the polar form of the midpoint of these two complex numbers? I think I'm correct in saying ... 2answers 62 views ### Help solve {{z}^{3}}=\overline{z} (z\in \mathbb{C}) [duplicate] Me and my friend try to solve$${{z}^{3}}=\overline{z}where z \in \mathbb{C}. My way to solve it was: \operatorname{cin}(\theta )=\cos(\theta)+\sin(\theta)i \begin{align} & z=r \... 3answers 37 views ### Electrical Engineering (complex numbers) Electrical Engineering (j=i=\sqrt{-1}):H_v(\omega)=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega CR}{1+Rj\omega C}$And we know that:$\omega_0=\frac{1}{RC}\Longleftrightarrow RC=\frac{1}{\...
The points $O$,$P$ and $Q$ in the complex plane represent the complex numbers $0+0i$, $4+2i$ and $3-i$ respectively. Find the exact length of $PQ$ and hence, or otherwise, show that triangle $OPQ$ is ...
The complex number $z$ is given by $z=-2+2i$ Find the modulus and argument of $z$ Write down the modulus and argument of $\frac{1}{z}$ Show on an Argand diagram the points A,B and C representing the ...