Complex Definition and 1000 Threads

Homework Statement Show that (A+B)*=A*+B* Homework Equations I think I am missing a property to prove this. The Attempt at a Solution This should be easier then I am making it out to be. But I seem to be missing one key property to do this. A*+B* is just A(ij)*+B(ij)* = Right hand side...

Say I will make the transformation from the ##z## plane to the ##w## plane. Moreover, I'll transform a region ##R## with boundary ##C## in the ##z## plane to something in the ##w## plane. Why is it that if I know the equations for ##C## then I can transform these and immediately know that ##R##...

Is correct to afirm that a solution of a quadratic equation or is a complex number or is a split complex number?

Homework Statement This is not for a mathematics unit but is part of an electrical question I'm trying to solve but I cannot solve this equation. The complex numbers Zp and Zr are both real and imaginary, whereas Xm is purely imaginary. Homework Equations Zp = (Xm*Zr)/(Xm+Zr) Zp =...

Some time ago I stumbled upon the integration ##\int \frac{dz}{z} ## along a line on the complex plane. I was confused because ##Ln(z)## is a multivalued function but apparently the way you do it is by only considering the principal branch from ##[-\pi,\pi]##. But I don't understand this at...

Just reviewing some QM again and I think I'm forgetting something basic. Just consider a qubit with basis {0, 1}. On the one hand I thought 0 and -0 are NOT the same state as demonstrated in interference experiments, but on the other hand the literature seems to say the state space is...

Homework Statement The following doesn't come from a textbook, and I am very uncertain whether it is true or false. Suppose that ##B \subseteq \mathbb{C}## is a convex set, and consider the set ##L_B := \{|b|: b \in B \}##. Homework EquationsThe Attempt at a Solution My question is, will...

If I define the complex number z = r exp(i θ) how z = uv, so, how to express u and v in terms of r and θ? u(r, θ) = ? v(r, θ) = ? And the inverse too: r(u, v) = ? θ(u, v) = ?

I was me asking why the complex numbers are defined how z = x + i y !? Is this definition the better definition or was chosen by chance? In mathematics, some things are defined by chance, for example: 0 is the multiplicative neutral element and your multiplicative inverse (0-) is the ∞. But, 1...

Hey there! I'm having some real trouble deciphering this complex resistor problem. I have heard of the Kirchhoff voltage and current rules and do know how to use them to solve some problems but I'm not sure how to apply them in this context, or if they are even used to solve this. As seen the...

$${x}^{2}+2i=0$$ $$\left(x-? \right)\left(x-? \right)=0$$ This should be easy but I couldn't get the factor

If the solution of the quadratic equation \frac{-b \pm \sqrt{b^2-4ac}}{2a} produces a new kind of number, the complex numbers a \pm i b so, the solution the cubic equation should to produce a new kind of number too, and the solution of the quartic equation too, etc...

I need to perform the following integration: ##I(s) = \frac{1}{2\pi i} \int_{\gamma}\text{d}z\ z^{-s} \frac{\text{d}\ln\mathcal{F(z)}}{\text{d}z}##, where ##\mathcal{F(z)}## is analytic everywhere on the complex plane except at the zeroes of the function. For the purpose of integration, the...

OK, so i took a course named "Oscillations and vibrations" We began the course with an "introduction" to complex numbers, basically we raced through them in like 3 classes, we talked about how to get complex roots, adding, multiplying, Cauchy-Riemann Conditions, Cauchy's integration Theorem...

Homework Statement First off i wasn't sure if i should put this in precalc or here so i just tossed a coin[/B] I must find the roots of the expression z^4 +4=0 (which I've seen repeatedly on the internet) Use it to factorize z^4 +4 into quadratic factors with real coefficients The answer is...

Dear PF Forum, I want to learn web programming, but there are specifics information that I need to know. What is the most famous database in web programming? My SQL? Is it true that PF Forum database is MySQL? If this is true, then the conclusion is MySQL can handle millions of post, hundreds...

I was wondering if scientists or mathematicians have any use for complex numbers involving negative roots of I as in i=(-1)^(1/2). but my question is more what would be (-1)^(-1/2)?

Notion of differentiability (analyticity) for function of complex variable is normally introduced and illustrated by comparison with function of single real variable. It is stated that there are infinite number of ways to approach any given point of complex plane where function is defined, not...

can you recommend a good book on complex analysis? I would like a book that can sharpen my skills in solving complex number problems through graphs and also improve the algebraic part like solving problems related to roots of unity etc. (I have studied calculus myself. I have done a lot of self...

I think I'm a bit rusty here, started with finding poles for $ \frac{1}{{z^4}+4} $, {z: |z-1| LE 2} 1) Out of interest, is there a complex equivalent of the rational roots test? The above function is obvious, but for a poly that has both real and complex roots? 2) I am using the exponential...

I am interested to find the length shown in red in the attached figure. I want this length as a function of d (shown in blue) and the angle θ. Then I will integrate this length to dθ from 0 to π/2. Firstly, I used the law of the triangle to determine the length s which when subtracted from the...

Homework Statement Hi all, I posted the image of the solution here. My questions concerns the evaluation of the Residue at z=i. Homework Equations None needed, all giving in the question -- simple algebraic mistake made is likely to be the problem here. The Attempt at a Solution [/B] We...

As you can see, it says that -110 (-1 to the tenth is just -1), multiplied by i, is somehow i. Everywhere I have looked, -1 times i is negative i, but this problem disagrees. Am I missing something?

Homework Statement I have uploaded necessary image(s) for the question I have successfully accomplished a, but I am not sure how to start b. Homework Equations The sum of the integral paths added up = the desired result. The Attempt at a Solution [/B] So we start with path CR And then go...

Homework Statement Computer the integral: Integral from 0 to infinity of (d(theta)/(5+4sin(theta)) Homework Equations integral 0 to 2pi (d(theta)/1+asin(theta)) = 2pi/(sqrt(1-a^2)) (-1<a<1) The Attempt at a Solution I've seen this integral be computed from 0 to 2pi, where the answer is 2pi/3...

Hi, an exercise asks to show that $ \int_{0,0}^{1,1} {z}^{*}\,dz $ depends on the path, using the 2 obvious rectangular paths. So I did: $ \int_{c} {z}^{*}\,dz = \int_{c}(x-iy) \,(dx+idy) = \int_{c}(xdx + ydy) + i\int_{c}(xdy - ydx) = \frac{1}{2}({x}^{2} + {y}^{2}) |_{c} + i(xy - yx)|_{c}...

Under some circumstances, whenever I call DEVCCG to diagonalize a general complex matrix, the program gets stuck inside and never returns. I do not even get out an error code so that I may continue with the rest of the program. I assume the iterative diagonalization inside the procedure does not...

Hi - in an example, I can't follow the working from one of the steps to the next, the 2 steps are: $... \sqrt{\frac{1}{2}\left(1-i\right)} = \sqrt{\frac{1}{\sqrt{2}}{e^{-i(\frac{\pi}{4}-2n\pi)}}}$ I can see they equate $ \frac{1-i}{\sqrt{2}} = e^{-i(\frac{\pi}{4}-2n\pi)}$, and I can see the $...

Hi An exercise asks to show $ \int_{a}^{b}f(z) \,dz = -\int_{b}^{a}f(z) \,dz $ I can remember this for real functions, something like $ G(x) = \int_{a}^{b}f(x) \,dx = G(b) - G(a), \therefore \int_{b}^{a}f(x) \,dx = G(a) - G(b) = -\int_{a}^{b}f(x) \,dx $ I have seen 2 approaches, either...

Homework Statement Problem and solution found here: http://homepages.math.uic.edu/~dcabrera/math417/summer2008/section57_59.pdf The question I am interested in is #1. In the solution, the instructor differentiates the series to get to: 2/(1-z)^3 = the series. If I want the Maclaurin series of...

How would I create a complex loan ammortization schedule for the following figures $390,000 Loan (3) payments of $10,000 each year on Jan 5th, July 5th and Oct 5th First Payment on July 5th 2015 Ammortized over 30yrs

Dear All, Please, I am trying to read in a data shown in array form but no luck. The data is 10 by 10 with each 10 by 10 depicted by a local array, for e.g. 0 0 0 0 0 0. A sample of the code I tried using is as shown below and the data is as attached. Please, any help or suggestion will be...

$a>1,b>1$ find the minimum value of $\dfrac {a^2}{b-1}+\dfrac {b^2}{a-1}$

Homework Statement Let 0 < r < 1. Show that from n=1 to n=∞ of Σ(r^ncos(n*theta)) = (rcos(theta)-r^2)/(1-2rcos(theta)+r^2) Hint. This is an example of the statement that sometimes the fastest path to a “real” fact is via complex numbers. Let z = reiθ. Then, since r = |z|, and 0 < r < 1, the...

Hi everyone, I have a dispersive wave packet of the form: ##\frac{1}{\sqrt{D^2 + 2i \frac{ct}{k_0}}} e^{-y^2/(D^2+2i\frac{ct}{k_0})}## The textbook says that the enlargement of the package, on the y direction, is: ##L=\frac{1}{D}\sqrt{D^4+4\left(\frac{ct}{k_0}\right)^2} ## However I have some...

Homework Statement With . Give an example, if it exists, of a non constant holomorphic function that is zero everywhere and has the form 1/n, where n € N. Homework Equations So.. This was in my Complex Analysis exam, and i have no idea what to do. I always seem to get stuck at these more...

1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...

I have an impedance tube and it gives me the magnitude (dB) and phase (phase) of a signaa. So does that mean the complex pressure at that point is simply the [Magnitude] +i[Phase Value]? Does I need to change the units at all?

Homework Statement This is an easy one, but keep in mind I'm kind of a newbie, anyway I can't figure out how to get the next formula... tan(z) = (tan(a)+i tanh(b))/(1 - i tan(a)tan(b)) Homework Equations This is the third part of an excercise, previous I proof the follow, -all using the...

Show that, for all real $$p$$ and $$m$$, $$e^{2mi\cot^{-1}(p)}\left(\dfrac{pi+1}{pi-1}\right)^m=1$$

Hi all, I was unsure where to put this thread as I read the main topic title in the topology/analysis forums and decided to post it here. I am looking for a chart/graph/website that helps me understand the basic terms such as: -neighborhoods -Boundary points -Singularity points - "Function is...

I neet to do a structured triangular mesh 2D of a complex geometry with FORTRAN Can someone help me!

I am currently learning how to work with Cauchy-Riemann equations. The equation is f(z) = 2x+ixy^2. My question: is u(x,y) = 2x or just x? At this link: http://www.math.mun.ca/~mkondra/coan/as3a.pdf in letter e) they say u(x,y) is equal to x. But I don't understand how that is possible. Is...

Dear all, I'd like to know what is the place/use of complex variables (and complex analysis) in classical mechanics. By the way, is there any? Thanks for your help. Best regards!

Homework Statement Find the image of the rectangle with four vertices A=0, B= pi*i, C= -1+pi*i, D = -1 under the function f(z)=e^x 2. The attempt at a solution So, the graph of the original points is obvious. Now I have to map them to the new function. Seems easy enough, but I am not getting...

Hi,I'm facing a problem finding the values of complex numbers, I'll put two examples then I'll explain the issue. ex1: (-e)^{iπ} , my answer is (-e)^{π^2±2mπ^2} The book answer is (-e)^{π^2} ex2: e^{2 arctanh(i)} , my answer is e^{[iπ/2±mπ/2]} = ie^{±mπ/2} The book answer is i...

The problem states, Show that: a) |e^(i*theta)| = 1. Now, the definition of e^(i*theta) makes this |cos(theta)+isin(theta)| If we choose any theta then this should be equal to 1. What can help me prove this? If I choose, say, pi/6 then it simplifies to |(sqrt(3))/2+i/2)| which doesn't seem to...

Use properties to show that: (question is in the attached picture) Now, it is my understanding that due to properties you can express (sqrt(5)-i) as the sqrt((sqrt(5))^2+(-1)^2) which equals sqrt(6). And (2zbar+5) can be represented as (2z+5). But this would be sqrt(6)*(2z+5) which is NOT...

Hi, I'm trying this summer to finish my mathematical methods book. I'm investigating right now the chapter of complex numbers, the end of the chapter has some applications in electricity and how can complex numbers make the work easier. The problem is that I didn't found it easier nor...

Hey, so I just have a quick question. I am trying to set a complex variable (in an array) as ##e^{i\alpha_1}## and the line I used in my code looks like this: hmajphasemix(2,2)=(cos(alpha1),sin(alpha1)) But the compiler is telling me that it "expects a right parenthesis" at this line. I'm...