Complex Definition and 1000 Threads

Hi. If a real integral between 2 values gives the area under that curve between those 2 values what does a complex integral give between 2 values ?

Homework Statement Prove that <v|0>=0 for all |v> ∈ V. Homework EquationsThe Attempt at a Solution This is a general inner product space. I break it up into 2 cases. Case 1: If |v> = 0, the proof is trivial due to inner space axiom stating <0|0> = 0. Case 2: If |v> =/= 0 then: I use <v|0>...

Thank you for opening my first thread. I have just began studying Physics at a university in Hungary and would like to get some information about Physics of Complex Systems and Plasma Phyics/Nuclear Fusion. In my life, finding a goal have always helped me work harder and focus on my studies...

The transformation T maps the plane onto itself by multiplication by a complex number. That is, there is a complex number C=a+ib such that for any point P(x,y), T(P) is the point corresponding to the complex number C⋅P. For a particular complex number C the transformation T takes the smaller...

Homework Statement Calculating the refraction of index (n) with given information. Homework Equations The Attempt at a Solution I tried to use Snell's Law but I have no idea about how I'm supposed to use it without angles. Instead, question gives distances. Need help very badly.

Homework Statement Let v ∈ V and c ∈ ℂ, with c ≠ 0. Prove that if cv = 0, then v = 0. Homework Equations Vector space axioms. The Attempt at a Solution Simple proof overall, but I have one major clarification question. v = 1v = (c^(-1)c)v = c^(-1) (cv) = c^(-1) 0 v = 0 My question is, in...

I appreciate the opportunity afforded by this forum to submit a question. I have struggled with the derivation shown in the attached picture. I am certainly unfamiliar with the concept used to include the arctan function in the encircled step. Would be highly appreciative of a prompt.wirefree

Homework Statement The problem is to sketch lines of constant u and v in the image plane for the function Log[(z+1)/(z-1)]. Homework Equations z=x+iy The Attempt at a Solution In order to do this I have to get the expression into u+iv form, so that I can read off and manipulate the u and v...

Homework Statement OABC is a square on an Argand diagram. O Represents 0, A represents -4 + 2i, B Represents z, C represents w and D is the point where the diagonals of the square meet. (There are two possible squares that meet this criteria) Find the complex number represented by C and D in...

This relates to z-transform causality, but I'll try to phrase it as a complex analysis question. Suppose I have a function ##X(z)## whose poles are all inside the unit circle, and which has the property \lim_{|z|\to\infty} \frac{X(z)}{z} = 0 Is that sufficient to guarantee that \frac{1}{2\pi...

As an undergraduate, which graduate-level course will prepare me better for grad school, Complex Analysis or Topology? I probably can't fit both into my schedule, but I can definitely fit one. I have already taken undergraduate complex analysis and I'm taking now undergraduate topology. My...

Hi everyone, Can you please assist with the following problem? The complex numbers z and w are such that for the real variable x, (x-z)(x-w)=ax2+bx+c for real a,b and c. By letting z=p+qi and w=r+si, prove that z and w must be conjugates of one another.So far, I have determined that a=1...

I know this is probably the least of my worries at the moment but my quantum textbook solves ##\frac{\mathrm{d}\phi (t) }{\mathrm{d} t}=\frac{iC}{h}\phi (t) ## as ##\phi (t) = e^{-i(\frac{C}{h})t}##. Is this not off by a sign? Its really bugging me.

Homework Statement A rope is wrapped through an angle θ about a horizontal pole (So for ex- ample, θ = 2π would imply the rope goes around one full time). The rope and the pole have a static friction coeffecient of μ, and the pole is of radius r. From one end of the rope hangs a mass m. How...

Attached image with problem. 2 Questions; 1. δn.|AB| = δx.|BC|.cosΘ becomes δn = δx.|BC|.cosΘ / |AB| This is dividing the entire "δx.|BC|.cosΘ" equation by |AB| or just the |BC| part? Is there a difference? 2. Half way down page, boxed equation. How does δx.|BC|.cosΘ / |AB| = δxcos^2Θ and...

Homework Statement I need to find the complex angle θ for: 2√(3)-2i in polar form. Homework EquationsThe Attempt at a Solution If I draw a complex plane, I can see that 2√3 on the real axis gives 0°, and -2i gives 3π/2 (270°), but it's incorrect. How can I find the complex angle of 2√(3)-2i...

(1+a)n = (1-a)n I tried following: (1+a)n = (1-a)n [(1+a)/(1-a)]n=1 but what can i do next?

Homework Statement Describe all the singularities of the function ##g(z)=\frac{z}{1-\cos{z}}## inside ##C## and calculate the integral ## \int_C \frac{z}{1-\cos{z}}dz, ## where ##C=\{z:|z|=1\}## and positively oriented. Homework Equations [/B] Residue theorem: Let C be a simple closed...

Homework Statement Let ##D={z : |z| <1}##. How many zeros (counted according to multiplicty) does the function ##f(z)=2z^4-2z^3+2z^2-2z+9## have in ##D##? Prove that you answer is correct. Homework Equations 3. The Attempt at a Solution [/B] The function has no zeros in ##D##, which can be...

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