Complex Definition and 1000 Threads

Homework Statement Find the equations of motion for the Lagrangian below: $$ L=\partial_\mu \phi^* \partial^\mu \phi - V( \phi,\phi^* ) $$ Where : $$ V( \phi,\phi^* )= m^2 \phi^* \phi + \lambda (\phi^* \phi)^2 $$Homework Equations Euler Lagrange equation: $$ \partial_\mu \dfrac {\partial L}...

Homework Statement i'm trying to calculate the charge operator for a complex scalar field. I've got the overal problem right but I'm confused about this: On page 18 of Peskin, it is written that the conserved current of a complex scalar field, associated with the transformation ##\phi...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding the first example on page 63 ... ... The the first...

I am searching for a shortcut in the calculation of a proof. The question is as follows: 2.12 Prove that: $$|z_1|+|z_2| = |\frac{z_1+z_2}{2}-u|+|\frac{z_1+z_2}{2}+u|$$ where $z_1,z_2$ are two complex numbers and $u=\sqrt{z_1z_2}$. I thought of showing that the squares of both sides of the...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding another aspect of the proof of Proposition 1.3...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding aspects of Proposition 1.3 ...Proposition 1.3 and its...

My residue is wrong. What is the solutions and the steps to achieve it ?

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 4: Complex Integration, Section 1.2 Smooth and Piecewise Smooth Paths ... I need help with some aspects of Example 1.3, Section 1.2, Chapter 4 ... Example 1.3, Section 1.2, Chapter 4...

Homework Statement My notes state the Lemma as shown above. I believe one of the underlying conditions is that the arc we integrate over must be +ve oriented (anti-clockwise) in the Upper and Lower half of the Complex Plane. However my notes doesn't mention whether or not the result holds...

Homework Statement For the expression $$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$ Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that: $$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...

Homework Statement Hi all, could someone help me run through my work for these 2 integrals and see if I'm in the right direction? I'm feeling rather unsure of my work. 1) Evaluate ##\oint _\Gamma Z^*dz## along an anticlockwise circle of radius R centered at z = 0 2) Calculate the contour...

Show geometrically that if |z|=1 then, $Im[z/(z+1)^2]=0$ I am unsure how to begin this problem. I have sketched out |z|=1 but can't work out how to sketch the Imaginary part of the question.

I have a circle with centre (-4,0) and radius 1. I need to draw the image of this object under the following mappings: a) w=e^(ipi)z b) w = 2z c) w = 2e^(ipi)z d) w = z + 2 + 2i I have managed to complete the question for a square and a rectangle as the points are easy to map as they are...

Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { } (This question is probably asked a lot but I could not find it so I'll just ask it myself.) Does the square root of negative numbers exist in the complex field? In other words is...

Homework Statement What is the argument of -4-3i, and -4+3i? Homework Equations tantheta=opposite/adjacent side The principle of argument is that the argument lies between -pi and pi (not including -pi). The Attempt at a Solution arg(-4-3i) = -pi + arctan(3/4) arg(-4+3i) = pi - arctan(3/4)...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects regarding an example in Palka's final remarks in Section 2.2 Limits of Functions ... Palka's final remarks in...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of a worked example in Palka's remarks in Section 2.2 Limits of Functions ... Palka's remarks in Section 2.2 which...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with some aspects of the proof of Lemma 2.4 ... Lemma 2.4 and its proof reads as follows: My questions are as follows: Question 1...

I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 1.6 The Topology of Complex Numbers ... I need help in fully understanding a remark by M&H ... made just after Example 1.22 ... Example...

Hi I was hoping some of you would give me a clue on how to solve this complex number task. z = (1 +(√3 /2) + i/2)^24 → x=(1 +(√3 /2), y= 1/2 I think there must be some nice looking way to solve it. My way was to calculate |z| which was equal to [√(3+2√3)]/2 → cosθ = x/|z|, sinθ= y/|z| After...

So say a wave is described by Acos(Φ), completely real. Then the to use Euler's Eq, we we say the wave is AeiΦ, which is expanded to Acos(Φ) + iAsin(Φ). We tell ourselves that we just ignore the imaginary part and only keep the real part. And if intensity is |AeiΦ|2, which is (Acos(Φ) +...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 2: The Rudiments of Plane Topology ... I need help with an aspect of Theorem 1.8 ... Theorem 1.8 (preceded by its "proof") reads as follows...

I asked my question at math.stackexchange with no reply as of yet, here's my question: https://math.stackexchange.com/questions/2448845/well-posedness-of-a-complex-pde Hope I could have some assistance here. [EDIT by moderator: Added copy of question here.][/color] I have the following PDE...

Homework Statement Solve the equation $$cos(\pi e^z) = 0$$Homework Equations I am not allowed to use the complex logarithm identities. $$ \cos z = \frac{e^{iz}+e^{-iz}}{2} $$ $$e^{i\theta}=\cos\theta+i \sin\theta$$ The Attempt at a Solution All I've gotten is $$\cos(\pi e^z)=0 \iff \pi...

Hi. If I have a complex number αe iα where α is complex what is the conjugate ? Usually I just replace i with -i but do i also take the conjugate of all α's ? Thanks

Homework Statement Homework Equations $$a^2-b^2=(a-b)(a+b)$$ The Attempt at a Solution $$a^2=\sqrt{1-x_2^2}\,\,\, ,\ \ b^2=\sqrt{1-x_1^2}$$ $$|a^2-b^2|=\left| \sqrt{1-x_2^2}-\sqrt{1-x_1^2} \right|=\left| \sqrt[4]{1-x_2^2} - \sqrt[4]{1-x_1^2} \right|\cdot\left| \sqrt[4]{1-x_2^2} +...

When solving problems, particularly in optics, it is often that we represent the wave-function as a complex number, and then take the real part of it to be the final solution, after we do our analysis. u(\vec{r},t)=Re\{U(\vec{r},t)\}=\frac{1}{2}\left(U+U^*\right) Here U is the complex form of...

Homework Statement Homework EquationsThe Attempt at a Solution I tried differentiating both sides of 3 and re-arranging it such that it started to look like equation 2, however i got stuck with 2 first order terms z' and couldn't find a way to manipulate it into a function z. I then tried...

Homework Statement Please see attachment. Homework Equations I don't know how to get the final product on the ones with the question marks (textbook answers written next to them). I've gotten to the last step (except for # 29 but don't mind that one, I haven't exhausted all ideas). I've...

if a is a complex number then sqrt(a^2)=? Is it is similar to Real Number? Help me please

Hi everyone, I'm reading about the solution of the telegrapher's equations (e.g. the generalities are here https://en.wikipedia.org/wiki/Telegrapher%27s_equations ). Supposing we are treating only time t and space z, this is a second order partial differential equation on an infinite domain of...

Hello everyone. Iam about to read a course in DSP and I can't get my head why complex exponentials are used as input signals that often? Is it just to simplify the math? If not, what exactly is the imaginary part of a complex exponential? Does it "do" anything special compared to a real valued...

For this, f and g are real functions, and k is a real constant. I have ##\psi = fe^{ikx}+ge^{ikx}## and I want to find ##\left|\psi \right|^2##. I went about this two different ways, and got two different answers, meaning I must be doing something wrong. Method 1: ##\psi =(f+g)e^{ikx}##...

Hello everyone. Iam reading about complex numbers at the moment ad Iam quite confused. I know how to use them but Iam not getting a real understanding of what they actually are :-( What exactly is the imaginary part of a complex number? I read that it could in example be phase... Thanks in...

Homework Statement In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation $$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$Homework Equations ##Co=\pi## ##\frac {ao} 2 = \pi## ##Cn=\frac j n## ##C_{-n}= \frac {-j} n ## ##an=0##...

Prop 5.11 from John M. Lee's "Introduction to Topological Manifolds":If K is a simplicial complex whose geometric realization is a 1-manifold, each vertex of K lies one exactly two edges. This proposition confuses me. If we look at the geometric realization of a simplex with two vertices, then...

Homework Statement Homework Equations $$(x-a)(x+a)=x^2-a^2$$ The Attempt at a Solution I have to express ##~\displaystyle x^2+16=f\left( \frac{x}{x-1} \right)## I guess it has to be ##~\displaystyle \left( \frac{x}{x-1} \right)^n-a~## or ##~\displaystyle \left( \frac{x}{x-1} \pm a...

@fresh_42 @FactChecker After thinking, I understood that the answer for this question might make the complex numbers comprehensible for me. My question in detail is as follow Let the equation of a sphere with center at the origin be ##Z1²+Z2²+Z3² = r²## where Z1 = a+ib, Z2 = c+id, Z3 = s+it...

Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?

Homework Statement Homework EquationsThe Attempt at a Solution I'm not sure how to even begin this problem. My notes mentioned something about a Mobius Transformation but that's not something that I've been taught, and certainly not something I'm familiar with. Any advice would be greatly...

Homework Statement Homework EquationsThe Attempt at a Solution I attempted to use the formula zj = xj + iyj to substitute both z's. Further simplification gave me (x1 + x2)cosθ + (y2 - y1)sinθ or, Re(z2 + z1)cosθ + Im(z2 - z1)sinθ. Is this a valid answer? Or are there any other identities...

$\tiny{s10.03.25}$ $\textsf{Write complex number in rectangular form}$ \begin{align*}\displaystyle z&=4\left[\cos\frac{7\pi}{4} + i\sin \frac{7\pi}{4} \right]\\ \end{align*} $\textit{ok from the unit circle: $\displaystyle\cos{\left(\frac{7\pi}{4}\right)}=\frac{\sqrt{2}}{2}$}\\$ $\textit{and...

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...