Complex Definition and 1000 Threads

hello all i'm trying to modal a complex material with matrix of material X and small spherical inclusion of material Y, i would like to have the ability to control the density of the inclusions and the surface properties between the material. does anyone know about a guide for the situation...

How is it possible to ignore the addition sign and imaginary number without contradicting fundamental Mathematics? I find it difficult to understand.

$\tiny{hcc8.11}$ $\textsf{Find product $(1+3i)(2-2i)$}\\$ $8 + 4i$ $\textsf{Then change each to complex form and find product. with DeMoine's Theorem}$ $\textit{ok looked at an example but ??}

I'm trying to derive the commutation relations of the raising and lowering operators for a complex scalar field and I had a question. Let's start with the commutation relations: $$[\varphi(\mathbf{x},t),\varphi(\mathbf{x}',t)]=0$$ $$[\Pi(\mathbf{x},t),\Pi(\mathbf{x}',t)]=0$$...

Related to the recent discussions on this forum about the potential for genetically engineering humans in the future, researchers from Stanford University recently published a fascinating article in the journal Cell, looking into the genetics of complex traits, like height, as well as the...

What book do MHB members regard as the best for a rigorous but clear and (moderately) easily understood introduction to complex analysis? (Note - would be good if the book had hints to solutions of exercise.) Peter

If I have a lagrangian which has terms of the form ##\Psi^{\dagger}_\mu \Psi^\mu## then I can decompose the n complex ##\Psi## fields into 2n real fields by ##\Psi_\mu = \eta_{2\mu+1} + i\eta_{2\mu}##. When I look at the lagrangian now it seems to have SO(2n) symmetry from mixing the 2n real...

I've been tasked with designing light shades for my companies new building. The current goal is to 3D print them, and include the company logo/name on them. The lights are for design purposes only, and aren't being used to illuminate the room. The hard part is, I want the logo/name clearly...

https://arxiv.org/pdf/1705.07188.pdf Equation 5 in this paper states that $$\frac{\partial F}{\partial p_i} = 2Re\left\lbrace\frac{\partial F}{\partial x}\frac{\partial x}{\partial p_i}\right\rbrace$$ Here, p_i stands for the i'th element of a vector of 'design parameters' \mathbf{p}. These...

Homework Statement Q:/ Find the complex form of Fourier series for the following periodic function whose definition in one period is given below then convert to real trigonometry also find f(0). f(t)=cos(t/2), notes: (T=2*pi) (L=pi) Homework Equations 1) f(t)=sum from -inf to +inf (Cn...

I wouldn't be surprised if I've posted in the wrong section because in fact the reason for posting is to get help naming this problem. That being the first step to knowing where to look for a solution. Newbie to the forum so open to advice. The problem: I have a complex histogram and a...

I'm looking for a good Complex book, but the options seem slim. I was thinking about Rudin's Real and Complex. My only reservation is that it is not structured like any other book I've seen. I've had advanced analysis and measure and integration theory, so rigour is not a concern. I saw Alfohr's...

This is spontaneous symmetry breaking problem. 1. Homework Statement Temperature is ##T=0##. For one component complex scalar field ##\phi##, non-relativistic Lagrangian can be written as $$ \mathcal{L}_{NR}=\varphi^* \Big( i\partial_t + \dfrac{\nabla^2}{2m} \Big)\varphi -...

Hi, My understanding is that when light (with some frequency and polarization) hits the interface between two media (each with some frequency-dependent material properties), the Fresnel equations apply. This tells us how much light reflects back versus refracts across the interface. I'm...

Hello all, Please look at the following: Solve the equation: \[\left | z \right |i+2z=\sqrt{3}\] where z is a complex number. I tried solving it, and did the following, which is for some reason wrong. I saw a correct solution. My question to you is why mine is not, i.e., where is my mistake...

Hi, I need to solve a system of first order partial differential equations with complex variables given by What software should I use for solving this problem..? The system includes 13 differential equations ...

Hello all, Three consecutive elements of a geometric series are: m-3i, 8+i, n+17i where n and m are real numbers. I need to find n and m. I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and...

Hello all, I am trying to find the algebraic representation of the following numbers: \[rcis(90^{\circ}+\theta )\] and \[rcis(90^{\circ}-\theta )\] The answers in the book are: \[-y+ix\] and \[y+ix\] respectively. I don't get it... In the first case, if I take 90 degrees (working with...

Hello all, Given a complex number: \[z=r(cos\theta +isin\theta )\] I wish to find the polar representation of: \[-z,-z\bar{}\] I know that the answer should be: \[rcis(180+\theta )\] and \[rcis(180-\theta )\] but I don't know how to get there. I suspect a trigonometric identity, but I...

Homework Statement Given that a complex number z and its conjugate z¯ satisfy the equation z¯z¯ + zi = -i +1. Find the values of z. Homework EquationsThe Attempt at a Solution

Homework Statement Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\ c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc## Homework Equations N/A The Attempt at a Solution I got stuck at the first part. if ##A## is positive...

Also, if you get the gist of what I am asking, I would greatly appreciate correction of my vocabulary.

Hello! If I have a real integral between ##-\infty## and ##+\infty## and the function to be integrated is holomorphic in the whole complex plane except for a finite number of points on the real line does it matter how I make the path around the poles on the real line? I.e. if I integrate on the...

Hello all, I wish to plot and following complex numbers on a plane, and to find out which shape will be created. I find it hard to figure out the first one, I believe that the others will follow more easily (the forth is also tricky). \[z_{1}=\frac{2}{i-1}\] \[z_{2}=-\bar{z_{1}}\]...

Homework Statement 1. V for C1 2. Let's Say "C" is Resistor then C1 = R1 etc. how to get Eq Resistance for this Circuit ?[/B]Homework Equations Faraday's Law, Kirchoff Law The Attempt at a Solution [/B] for no 1 :I just want to make sure. point A is transfering negative charge to C2 and...

Homework Statement Just problem 19C. Homework Equations P=IV=Ieiwt*Veiwt. T The Attempt at a Solution P = IVe2iwt=IVcos(2wt). What did I do wrong?

Hiya all, I need your assistance with the following problem: A) Show that the equation \[z^{2}+i\bar{z}=(-2)\] has only two imaginary solutions. B) If Z1 and Z2 are the solutions, draw a rectangle which has the following vertices: Z1+3 , Z2+3 , Z1+i , Z2+i I do not know how to even...

Hello everyone, I have a complex number problem that i would greatly appreciate some help with. Thanks in advance to anyone offering their time to make a contribution. Q) Write the following in polar form: I have attempted the question (please see my working below) and have been advised that i...

Why does every subfield of Complex number have a copy of rational numbers ? Here's my proof, Let ##F## is a subield of ##\Bbb C##. I can assume that ##0, 1 \in F##. Lets say a number ##p \in F##, then ##1/p \ p \ne 0## and ##-p## must be in ##F##. Now since ##F## is subfield of ##\Bbb C##...

<Moderator note: Merger of two threads on the topic.> Hello! I am reading some basic stuff on complex integrals using branch cuts and i found the problem in the attachment. I am not sure I understand why the branch cut is along ##R^+##. I thought that branch cut is, loosely speaking, a line...

Hello! I found a proof in my physics books and at a step it says that: ##\int_m^\infty\sqrt{x^2-m^2}e^{-ixt}dx \sim_{t \to \infty} e^{-imt}##. Any advice on how to prove this?

Hi! If a holomorphic function ##f:G \to C##, where ##G## is a region in the complex plane is equal to zero for all values ##z## in a disk ##D_{[z_0,r]}##, inside ##G##, is it zero everywhere in the region G? And if this is true, does it mean that if an entire function is zero in a disk, it is...

Hello! I see that all theorems in complex analysis are talking about a function in a region of the complex plane. A region is defined as an open, connected set. If I am not wrong, the real line, based on this definition, is a region. I am a bit confused why there are so many properties of the...

Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof...

Homework Statement Suppose f is entire and there exist constants a and b such that ##|f(z)| \le a|z|+b## for all ##z \in C##. Prove that f is a polynomial of degree at most 1. Homework EquationsThe Attempt at a Solution We have that for any ##z \neq 0##, ##\frac{|f(z)|}{a|z|} \le b##. So if we...

Hello! I started learning about complex analysis and I am a bit confused about integration. I understand that if we take different paths for the same function, the value on the integral is different, depending on the path. But if we use the antiderivative...

It's not a homework question. I just thought up a method of finding answers to problems where a number is raised to a complex number and I need to know if I am right. If we have to find e^(i), can we do it by; first squaring it to get, e^(-1) which is 1/e and then taking its square root to get...

Hi, My teacher tasked me with a complex waveform question, i have looked for some time to find out how to tackle these, but i still do not know where to begin. Any help would be greatly appreciated, not look for an answer just a method. i=12sin(40*\pi t) + 4sin(120* \pi t - /3\pi) + 2sin(200...

I am a undergraduate student of engineering and I'm planning to go for Master's in physics department. I've watched some websites of research faculty or groups and I think (correct me if I'm wrong ;) there are main theoretical and experimental fields of these: - Elementary particles - Condensed...

Homework Statement exp(z)=-4+3i, find z in x+iy form Homework Equations See attached image. The Attempt at a Solution See attached image. exp(z)=exp(x+iy)=exp(x)*exp(iy)=exp(x)*[cos(y)+isin(y)] ... y=inv(tan(-3/4)=-.6432 ... mag(-4+3i)=5, x= ln (5)..exp(ln(5))=5 ...

Homework Statement We are given Z, and are asked to find the magnitude of the expression. See attached picture(s) Homework Equations See attached pictures(s) The Attempt at a Solution When I solved it on the exam, I did it the long way using De Moivre's theorem. I ended up making a few sign...

Homework Statement $$z^2 + z|z| + |z|^2=0$$ The locus of ##z## represents- a) Circle b) Ellipse c) Pair of Straight Lines d) None of these Homework Equations ##z\bar{z} = |z|^2## The Attempt at a Solution Let ##z = r(cosx + isinx)## Using this in the given equation ##r^2(cos2x + isin2x) +...

Hi everyone. I was looking at complex numbers, eulers formula and the unit circle in the complex plane. Unfortunately I can't figure out what the unit circle is used for. As far as I have understood: All complex numbers with an absolut value of 1 are lying on the circle. But what about...

Hi guys, I need some help on sketching graph complex functions such as ( 2x-⅜+¾e^-2x). Can someone please help me on sketching a graph like the one that I mentioned above. Is there any useful videos or website I can use. And please let me know if there are any good tips to get accurate...

I am trying to do the following integral: $$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)}.$$ Wolfram alpha - http://www.wolframalpha.com/input/?i=integrate+(cos(x))^(1/2)+dx+from+x=pi+to+3pi gives me $$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)} = 4f E(2) = 2.39628f + 2.39628if,$$ where E is the...

Hello everyone. Iam trying to get my head around the difference between real and complex numbers, but Iam having a hard time... I read that the difference is that a complex signal contains phase information. If I look at a real signal --> x(t) = Acos(wt + Θ) and compare...

Homework Statement $$\int_{-\infty}^\infty \space \frac{cos(2x)}{x-3i}dx$$ Homework EquationsThe Attempt at a Solution $$\int_{-R}^R \space \frac{e^{2ix}}{x-3i}dx + \int_{C_R} \space \frac{e^{2iz}}{z-3i}dz = 2\pi i\sum\space res \space f(z)$$ Then using Jordan's Lemma, as ##R\to\infty## the...

Homework Statement Compute the Integral: ##\int_{-\infty}^\infty \space \frac{e^{-2ix}}{x^2+4}dx## Homework Equations ##\int_C \space f(z) = 2\pi i \sum \space res \space f(z)## The Attempt at a Solution At first I tried doing this using a bounded integral but couldn't seem to get the right...

So the question is show that $$S=\left\{ \begin{pmatrix} a & b\\ -b & a \end{pmatrix} :a,b \in \Bbb{R} ,\text{ not both zero}\right\}$$ is isomorphic to $\Bbb{C}^*$, which is a non-zero complex number considered as a group under multiplication So I've shown that it is a group homomorphism by...

Homework Statement STATEMENT ##\hat{H}=\int \frac{d^3k}{(2\pi)^2}w_k(\hat{a^+(k)}\hat{a(k)} + \hat{b^{+}(k)}\hat{b(k)})## where ##w_k=\sqrt{{k}.{k}+m^2}## The only non vanishing commutation relations of the creation and annihilation operators are: ## [\alpha(k),\alpha^{+}(p)] =(2\pi)^3...