Complex Definition and 1000 Threads

I found the above while going through my textbook, where the textbook was trying to explain that the principal value of the product of two complex numbers raised to an exponent is not necessarily equivalent to the product of the two complex number each raised to the same exponent first. Based...

I would like to prove that ##\displaystyle{\int dx'\ \frac{1}{\sqrt{AB}}\exp\bigg[i\frac{(x''-x')^{2}}{A}\bigg]\exp\bigg[i\frac{(x'-x)^{2}}{B}\bigg]=\frac{1}{\sqrt{A+B}}\exp\bigg[i\frac{(x''-x)^{2}}{A+B}\bigg]}## Is there an easy way to do this integration that does not involve squaring the...

Hello, I've heard about "H.M." and similar patients that lost the ability to create declarative memories, but can still create non-declarative memories. I was wondering, since one in this condition can for example, develop a habit of drawing complex shapes, and not know they did it, can they...

Given that we can satisfy the wave equation with a simple sine & cosine wave function (the real part of the complex wave function) in classical mechanics, why do we use the complex wave function in EM theories? In QM it is obvious that it gets more mathematically more consistent. Out of curiosity.

Hey guys, I got the following derivation for some physical stuff (the derivation itself is just math) http://thesis.library.caltech.edu/5215/12/12appendixD.pdf I understand everything until D.8. After D.7 they get the eigenvalue and eigenvectors from ε. The text says that my δx(t) gets aligned...

Let's say you have a good personal computer with these characteristics: i7-6700K, 32GB RAM, 256GB SSD, 2 x GTX 970 4GB And let's assume you can run your algorithms, software... for 24 hours. I wonder, if I would be calculating the orbital structure of an hydrogen atom with many electrons...

When working in the complex domain (##z = x + iy##), how does one write the equation of a line? I have attached a problem I was working on (and have the solution), but am curious as to why the definition of a line is given by ##ax + by = c##. Are not ##x## and ##y## also variables that take on...

I've attached two equivalent complex equations, where one is written as a polynomial with 7 terms and the other is the factored form. I was just wondering how one can immediately write down the factored form based on the equation with 7 terms? Is there anything obvious (e.g. coefficient 1) or...

Hello all, I am trying to bring this: (p \iff q ) \implies r into a CNF form. I have started with the logical equivalences: (p \implies q) = \lnot p\lor q (p \iff q) = (p \land q)\lor (\lnot p \land \lnot q) and then I have applied De Morgan's rules and the distribution rules, but...

I am trying to find the limit of ## \frac {z^2 + i}{z^4 - 1} ## as ## z ## approaches ##i##. I've broken the solution down to: ##\frac {(z + \sqrt{i})(z - \sqrt{i})}{(z+1)(z-1)(z+i)(z-i)} ## but this does not seem to get me anywhere. The solution says ## -0.5 ## but I don't quite understand how...

Homework Statement $$ \left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right | $$ Where z and w are complex numbers not equal to zero. 2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...

Homework Statement Prove that the Noether charge ##Q=\frac{i}{2}\int\ d^{3}x\ (\phi^{*}\pi^{*}-\phi\pi)## for a complex scalar field (governed by the Klein-Gordon action) is a constant in time. Homework Equations ##\pi=\dot{\phi}^{*}## The Attempt at a Solution...

I find it difficult to believe that the canonical commutation relations for a complex scalar field are of the form ##[\phi(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## ##[\phi^{*}(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## This seems to imply that the two...

Homework Statement Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms. Degree 4: No real zeros, complex zeros of 1+i and 2-3i Homework Equations (-b±√b^2-4ac)/2a The Attempt at a Solution I want...

Homework Statement Consider 3 nonzero complex numbers $$z_1,z_2,z_3$$ each satisfying $$z^2=i \bar{z}$$. We are supposed to find $$z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$$. The answers- 0, purely imaginary , purely real respectively. Homework EquationsThe Attempt at a Solution I have...

Complex numbers If z=rcis(theta) FIND: |iz^2| I am confused about how I incorporate the i into the absolute value. I can't remember what it means. Please help and show exactly how I complete the workings. I can easily find the absolute value of z^2 I just really don't understand how to put the...

Hi, Just curious if someone knows of any Characteristic class used to determine if a manifold allows a Complex structure? It seems strange that Complex Space C^n is topologically Identical to R^{2n} yet I believe not all R^{2n}s ( if any) allow Complex structures. Thanks for any comments, refs...

Hi. I'm confused about the action of the complex conjugate operator and time reversal operator on kets. I know K(a |α > ) = a* K | α > but what is the action of K on | α > where K is the complex conjugation operator ? What is the action of the time reversal operator Θ on a ket , ie. what is Θ...

I am with a query about cw complex. I was thinking if is possible exist a cw complex without being of Hausdorff space. Because i was thinking that when you do a cell decomposition of a space (without being of Hausdorff) you do not obtain a 0-cell. If can exist a cw complex with space without...

Homework Statement ask to find all the values in z to the equation to be true[/B]Homework Equations [/B]The Attempt at a Solution this is my atemp of solution i don't know what else do, because the problem ask for z values well i must add that i am thinking about x=0 and y= pi/2 will solve...

Ok. I will try to do my best to explain you what is my doubt. I'm not a native English Speaker and the book I was reading is not in English Lang, but I've translated it to English. In a spin-1/2 Stern Gerlach-like experiment, we can express the ket representing the spin component of the...

Homework Statement Write this complex number in the form "a+bi" a) cos(-pi/3) + i*sin(-pi/3) b) 2√2(cos(-5pi/6)+i*sin(-5pi/6)) Homework Equations my only problem is that I am getting + instead of - on the cosinus side.(real number) The Attempt at a Solution a) pi/3 in the unit circle is 1/2...

Homework Statement Homework Equations Theta = arctan (y/x) The Attempt at a Solution Hopefully this is the right section to post in, but I am a bit confused with complex numbers. I am working on the problems above and I just wanted to make sure I am doing each part correctly. I think A...

Homework Statement Write the given numbers in the polar form ##re^{i\theta}##. ## \frac {2i} {(3e^{4+i})} ## Homework Equations ## z = re^(i\theta) ## ## \theta = Arg(z) ## ## r = |z| = \sqrt { x^2 + y^2 } ## The Attempt at a Solution I'm not really sure how to go about the exponential...

To give a bit of context, I am doing my final year university project on micro-mechanical interactions between an AFM probe and a sample surface. I do not have notes for a system this complicated, as we always considered our systems to be rigid bodies. I was always relatively clueless at...

Homework Statement To give a bit of context, I am doing my final year university project on micro-mechanical interactions between an AFM probe and a sample surface. I do not have notes for a system this complicated, as we always considered our systems to be rigid bodies. I was always...

need a hand with a revision question, I don't quite understand how to go about solving it question is attached below

So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its...

C1 1. Homework Statement : Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz where the contour C = C1 + C2. C1 is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C2 is the line segment from the origin to B...

Hi everyone Can anyone help me understanding the physical meaning for the complex dielectric constant? Assuming a electromagnetic wave from air to a conductor, the following equation is valid R= ((n-1)2+k2)/((n+1)2+k2) where K is the extinction coefficient (the complex part of the complex...

Hey all, I need the complex version of the sigmoid function in standard form, that is to say $$f(\alpha) =\frac{1}{1+e^{-\alpha}} , \hspace{2mm}\alpha = a+bi , \hspace{2mm} \mathbb{C} \to \mathbb{C}$$ in the simplified form: $$f = m+ni$$ but found this challenging, for some reason i assumed...

Homework Statement limit (5/(2+(9+x)^(0.5))^(cosecx) x-->0 attempt: tried applying lim (1+x)^(1/x) = e. x->0 couldn't get anywhere.

Homework Statement If modulus of z=x+ iy(a complex number) is 1 I.e |z|=1 then find the argument of z/(1+z)^2 Homework Equations argument of z = tan inverse (y/x) where z=x+iy modulus of z is |z|=root(x^2+y^2) The Attempt at a Solution z/(1+2z+z^2) = x+iy / 1+2(x+iy)+( x+iy)2 ...

Question 1: (a) Show that the complex number i is a root of the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 (b) Find the other roots of this equation Work: Well, I thought about factoring the equation into (x^2 + ...) (x^2+...) but I couldn't do it. Is there a method for that? Anyways the reason I...

Dear Physics Forum friends, I will be doing a reading course in the complex analysis starting on this Fall Semester. The assigned book is Rudin's Real and Complex Analysis. From my understanding, Rudin treats complex analysis very elegantly, but very terse. I am curious if you could suggest...

Homework Statement calculate ## \int_{-\infty}^{\infty}{\frac{2}{1+x^2} e^{-ikx}dx}##, where k is any positive number Homework EquationsThe Attempt at a Solution So first consider the closed contour ##I= \int{\frac{2}{1+z^2} e^{-ikz}dx}## We can choose the contour to be along the real axis ##...

I am dealing with a 3x4 complex matrix M that relates a vector d to another vetor c. That is: c = [M]*d d is 4x1 and c is 3x1. I want to introduce a new line (constraint) into M, say d(1) = d(2). However, I would like to only apply the constraint to the real or only the imaginary parts. Is...

I've been thinking about notions like the following: "How far can one be from the nearest road while in a particular country." "What's the 'maximum thickness' of a subset of \mathbb{R}^n?" "What mountain range has the biggest circular region entirely within it?" These sorts of questions lead...

Hi all, I have spent a couple of hours on this perplexing question. Solve the simultaneous equations: z = w + 3i + 2 and z2 - iw + 5 - 2i = 0 giving z and w in the form (x + yi) where x and y are real. I tried various methods, all to no avail. I have substituted z into z2 , I got the wrong...

What is the name for a toplogical space that is everywhere "locally like a simplical complex" in that every point has at least one neighbourhood which is either a topological manifold, or can be countably decomposed by surgery into a set of topological manifolds which intersect along...

Hi, I have a question regarding corollary 2.3. in the uploaded image. it looks very trivial to me becauese Cauchy's theorem states "if f(z) is holomorphic, its closed loop integral will be always 0". Is this what the author is trying to say? what's the necesseity of the larger disk D' at here...

What kind of steps do you use when you need to solve a complex physics problem? What kind of questiono do you ask yourself? Richard Feynman said that you have to write down the problem and then think hard, but what thoughts and questions can help yourself find a creative solution? Is creativity...

The problem The following equation ##z^4-2z^3+12z^2-14z+35=0## has a root with the real component = 1. What are the other solutions? The attempt This means that solutions are ##z = 1 \pm bi##and the factors are ##(z-(1-bi))(z-(1+bi)) ## and thus ## (z-(1-bi))(z-(1+bi)) =...

Homework Statement The integral I want to solve is $$ D(x) = \frac{-i}{8\pi^2}\int dr\,d\theta \frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}}r^2\sin\theta$$ which I've reduced to $$ D(x) = \frac{-i}{4\pi x}\int dr \frac{r\sin(rx)}{\sqrt{r^2+m^2}} $$ by integrating over ##\theta##. However, I...

Hi everyone! I have been perusing PF for some time, and what's struck me is how little discussion there has about research in complex systems & complexity, such as the research conducted in places like the Santa Fe Institute or the Center for the Study of Complex Systems at the University of...

Homework Statement Given the circuit of sinusoidal current (attachment 1) with given data: \underline{Z_3}=200(3-j4)\Omega, \underline{Z_4}=100(3+j20)\Omega, \underline{Z_5}=100(3+j4)\Omega, \underline{Z}=100(2+j5)\Omega, \underline{I_{g2}}=-10(2-j)mA After the switch is closed, the increment...

Homework Statement Homework EquationsThe Attempt at a Solution There is 0.2 mol of Cl- ions so therefore this leaves us with a 1:1 ratio for the chloride ion to chromium ion in the complex ion. This leaves us with option B or D. However, I am not sure how to choose between them. Can someone...

I am looking to numerically solve the (complex) Time Domain Ginzburg Landau Equation. I wish to write a python simulator to observe the nucleation of fluxons over a square 2D superconductor domain (eventually 3D, cubic domain). I am using a fourth order Runge Kutta solver for this which I made...

If I have 2 complex numbers, A and B, what is the correct way to evaluate this expression: ## E = AB - B^*A^*## I was under the impression that when taking the product of complex numbers, you always conjugate one factor, but in this instance, it is quite important which one is conjugated, no...

I know there's a similar post, but i didn't understand it. Why the derivative respect to t in terms of the complex conjugate of ψ is: instead of being the original S.E in terms of ψ* or the equation in terms of ψ with the signs swapped