Complex plane Definition and 128 Threads

Homework Statement I want to find the value of the integral: ∫cos(x)/((x+a)2+1) dx from ]-∞;∞[ Homework Equations Residue theorem The Attempt at a Solution My question is seeking more a conceptual understanding of why transforming to the complex plane works. According to...

Homework Statement Describe the locus and determine the Cartesian Equation of: \left|z-3-5i\right|= 2 Homework Equations \left|z-C\right|= r -----> formula for a circle on complex plane Where C = the centre z = the moving point (locus) (x-h)^{2}+(y-k)^{2}=r^{2} -----> Formula...

Find the locus defined by |z-2|-|z+2|=3 The given example rewrites the left hand side as: \sqrt{(x-2)^2+y^2}-\sqrt{(x+2)^2+y^2} 1) When they rewrite it as that, they square it and square root it right? Why isn't it squaring the whole expression and rooting it (isn't that the rule?)...

Homework Statement Find the 6th complex roots of √3 + i. Homework Equations z^6=2(cos(π/6)+isin(π/6)) r^6=2, r=2^1/6 6θ=π/6+2kπ, θ=π/36+kπ/3 The Attempt at a Solution When k=0, z = 2^1/6(cos(π/36)+isin(π/36)), When k=1, z = 2^1/6(cos(13π/36)+isin(13π/36)), When k=2, z =...

Homework Statement The following sum of integrals has integrals that are both integrated over straight lines in the complex plane. Deform the contours back to the origin and avoid the singularity at x = infinity to prove the integral formula, \int_{ - {x_0}}^\infty {(x + {x_0}){e^{ -...

Hey all, I have a vector field described by a complex potential function (so I have potential lines and streamlines). I am looking for a way to express its curvature at every point, but I can't find such a formula in my books. I have searched in wikipedia and I read that the way to define it...

Area of Region Bounded by the locus of $z$ which satisfy the equation \displaystyle \arg \left(\frac{z+5i}{z-5i}\right) = \pm \frac{\pi}{4} is

Consider the region A in the complex plane that consists of all points z such that both \frac{z}{40} and \frac{40}{\overline{z}} have real and imaginary parts between 0 and 1, inclusive. What is the integer that is nearest the area of A? Let z = a + bi and \overline{z} = a - bi a = real part...

Homework Statement Sketch: {z: \pi?4 < Arg z ≤ \pi} Homework Equations The Attempt at a Solution Is it right to assume z0 = 0 ; a = a (radius = a) ; and taking \alpha = \pi/4 ; \beta = \pi And now in order to sketch the problem after setting up the complex plane is it correct...

How can I draw a rectangle oriented clockwise on the complex plane with vertices on (0,0), (0,4), (10,4), and (10,0)? I am guessing the tikz package needs to be used but I am not skilled in making pictures.

Homework Statement |2z -1|\geq|z + i| The Attempt at a Solution The problem I have with this one is the 2z, I just need a clue on how to go about centering this one. If it were just |z - 1|; z_{0} would be 1.

So this is the problem as written and I'm totally lost. Any help or explanation would be greatly appreciated. "Viewing ℂ=ℝ2 , we can identify the complex numbers z = a+bi and w=c+di with the vectors (a,b) and (c,d) in R2 , respectively. Then we can form their dot product...

Homework Statement I do not have specific problem, I am struggling in my complex variables class and I think a large part of it is because I struggle at sketching regions in ℂ. For instance let z=x+ I full understand what |z|< 1 looks like and all that (punctured disk, things in that...

Can someone please describe to me how Euclidean Geometry is connected to the complex plane? Angles preservations, distance, Mobius Transformations, isometries, anything would be nice. Also, how can hyperbolic geometry be described with complex numbers?

The motion of a point P in the complex plane is defined by the principal root of z^5= (1+ t)^i a)find z(t) b)Show that P is undergoing a circular motion. Find the velocity and acceleration as a function of time I'm pretty sure I know how to do b but I don't really understand the...

Let A be a non-empty subset of the complex plane and let b ∈ ℂ be an arbitrary point not in A. Now define d(A,b) := inf{|z-b| : z ∈ A}. Show that if A is closed, then there is an a ∈ A such that d(A,b) = |a-b|. Ok so basically what I did was begin by choosing some arbitrary element of A and...

I'm not sure I understand the complex plane very well. For the cartesian plane, or other planes such as polar, points are plotted by a function. One value of x coresponds to a value of y. (or r to theta, or whatever.) The complex plane isn't a plot of functions, just of a single number...

Homework Statement Use suitable contours in the complex plane and the residue theorem to show that integral from -infinity to +infinity of [1/(1+(x^4))] dx=pi/(sqrt(2)) Fix R > 1, and consider the counterclockwise-oriented contour C consisting of the upper half circle of radius R...

Hi there, I have to prove this two sentences . I think I've solved the first, but I'm quiet stuck with the second. The first says: 1) Demonstrate that the equation of a line or a circumference in the complex plane can be written this way: \alpha z . \bar{z}+\beta z+\bar{\beta}...

Hello, There is a definition of simple-connectedness for a region R of the complex plane C that states that a region R is simply-connected in C if the complement of the region in the Riemann Sphere is connected. I don't know if I'm missing something; I guess we are actually consider...

Homework Statement if z= sin (omega) find an expression for omega as a function of z that can be used to evaluate all possible values of sin^(-1) (3). Plot these values on the complex plane The Attempt at a Solution z= sin (omega) 3= sin (omega) I don't know how to proceed from...

I am reading Visual Complex Analysis by Dr. Tristan Needham and am hung up on some of the geometrical concepts. In particular, I am having trouble with ideas involving the geometric properties of numbers like: \frac{z-a}{z-b} Note: I am still in the first and second chapters, which deal...

Homework Statement sketch on the complex plane the region where the following two power series both converge 1) sigma from n=0 to infinity [(z-1)^n]/[n^2] 2) sigma from n=0 to infinity [((n!)^2)((z+4i)^n)]/[2n]! The Attempt at a Solution R=lim as n tends to infinity...

Homework Statement Let z= x + yi be a complex number. and f(z) = u + vi a complex function. As: u = sinx\astcoshy v= cosx\astsinhy And if z has a trajectory shown in the attached image. What would be the trajectory of the point (u,v) ?

Hey guys so I was thinking about how to extend the Complex Plane out to a third dimension and I started reading the whole tidbit about Quaternions and their mechanics when I realized that I want to propose a whole new question. Now please feel free to prove me wrong if you can answer it because...

Homework Statement Hi everyone. I must show that if f is a continuous function over the complex plane, with limit as z tends to infinity = 0, then f is in fact bounded. The Attempt at a Solution Since f is continuous and lim z --> infinity f(z) = 0, by definition of limit at infinity I know...

I have been playing with the FFT and graphs. The easiest example I could think of for a transform was the top hat function (ie 0,0,0,0,0...1,1,1...0,0,0,0,0). When I transform this from the time domain to the frequency domain, it returns a sinc function when I take the absolute value squared of...

Homework Statement Graph the following inequality in the complex plane: |1 - z| < 1 2. The attempt at a solution In order to graph the inequality I need to get the left side in the form |z - ...| |1 - z| < 1 |(-1)z + 1| < 1 |-1(z - 1)| < 1 |-1||z - 1| < 1 (1)|z - 1| < 1 |z - 1| < 1...

Hi, so my question is the subject line. In the multiply connected domain |z|>0, does the function f(z) = e^z/z^3 have an antiderivative? I'm learning from Brown and Churchill, and they have a theroem on pg. 142 that leads me to believe it does. I don't remember what my prof said about this...

Homework Statement Problem 1. Create a program to display a complex number (or a list of complex numbers) as an arrow (or arrows) on the complex plane. i know RandomComplex[] will give me a random complex number, and i know RandomComplex[{1 - I, 1 + I}, 5, WorkingPrecision ->...

My book just gives me what each individual piece is but doesn't explain anything.

Homework Statement Show that the function f(z) = Re(z) + Im(z) is continuous in the entire complex plane. Homework Equations The Attempt at a Solution I know that to prove f(z) is a continuous function i have to show that it is continuous at each part of its domain. I take...

Homework Statement Draw |z| on a complex plane, where z = -3+4i Homework Equations N/A The Attempt at a Solution [PLAIN]http://img530.imageshack.us/img530/1786/aaakr.jpg Can anyone please tell me which answer is correct? Both of them have a moduli of 5. So should the circle...

Homework Statement Sketch the set of complex numbers z for which the following is true: arg[(z+i)/(z-1)] = \pi/2 Homework Equations if z=a+bi then arg(z) = arctan(b/a) [1] and if Z and W are complex numbers then arg(Z/W) = arg(Z) - arg(W) [2] The Attempt at a Solution using eq. [2] i...

Homework Statement \left\{ e^{n r \pi i}: n \in \textbf{Z} \right\} , r \in \textbf{Q} I'm trying to show that this set is finite. Homework Equations The Attempt at a Solution Other than the fact that these points lie on the unit circle in the complex plane, I'm not sure...

Definition of "compactness" in the EXTENDED complex plane? How does one define a compact set in the extended complex plane \mathbb C^* = \mathbb C \cup \{ \infty \}? "Closed and bounded" doesn't really make sense anymore, as I'm assuming it's permissible for a compact set to contain the point...

Homework Statement I know following that |z| = 1 where z \in \mathbb{C} is the definition of unit circle in the complex plane. then if the exist another complex number c which lies within the distance r from z then distance from the two numbers kan be discribe as |z-c| = r If...

given a finite polynomial a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+...+a_{n}x^{n} =P(x) is there a theorem or similar to ensure that P(x) has NO roots on the left of complex plane defined by Re(x<0) ??

suppose i have a real function f=f(x) this function is smooth everywhere on the real line for example, f=e^x. The problem is, is the continuation of the function into the complex plane unique? if so, does it hold that f(z)=f(z*)*?

Homework Statement I need to solve \int_L \bar{z}-1 where L is the line from 1 to 1+2i. Homework Equations The Attempt at a Solution I know that I need to set z equal to the equation of the line and then integrate, but in this case I'm not sure how to express the equation of...

Homework Statement The function f(z) = (1-z2)1/2 of the complex variable z is defined to be real and positive on the real axis in the range -1 < x < 1. Using cuts running along the real axis for 1 < x < infinity and -infinity < x < -1, show how f(z) is made single-valued and evaluate...

Homework Statement [/b] Graph the locus represented by the following. \left|z+2i\right| + \left|z-2i\right| = 6 Homework Equations The Attempt at a Solution z = x + iy so z-2i = x + (y-2)i and z+2i = x + (y-2)i So I have: sqrt(x^2 + (y-2)^2) + sqrt(x^2 + (y+2)^2) = 6...

Hi, I have one serious problem while solving rayleigh equation using blasius profile, in which so as to remove the singularity the intergration contour is defined in a complex plane. 4th order runge kutta is used but if the step size( h) is is in complex, it is giving some error. would anyone...

How does one express mathematically the fact that: if we complex-conjugated everything (switch i to -i (j to -j etc. in hypercomplex numbers) in all the definitions, theorems, functions, variables, exercises, jokes ;-)) in the mathematical literature the statements would still be true?

Homework Statement For a function f(z)= [e^(2*pi*i*a*z)] / [1 + z^2] I need to find the order of the poles at i and -i. (I'm pretty sure these are the only poles.) Homework Equations The Attempt at a Solution I'm not totally clear on how I go about finding the orders. I have a...

what is the relationship (if any) of the following statement - A function has ALL the zeros on the line (complex plane) Re (z) = A for some Real A - A function has ALL the zeros on the unit circle defined by |z| \le 1 i think there is a transformation of coordinates so the line Re...

Homework Statement Graph the following in the complex plane {zϵC: (6+i)z + (6-i)zbar + 5 = 0} Homework Equations z=x+iy zbar=x-iy The Attempt at a Solution Substituting the equations gives 2(6x-y) + 5 = 0 => y = 6x + (5/2) But that's a line in R^2. The imaginary parts...

Suppose you're trying to provide a branch cut in \mathbb{C} that will define a single-value branch of f(z) = \log(z - z_0). I don't know where to begin. Can someone help explain this concept to me?

Hi, I know this one is stupid, but i am still confused. why e^(iy) = cosy + i siny? thank you.

Homework Statement Identify and sketch the region in the complex plane satisfying | \frac{2 z - 1}{z + i} | \geq 1 Homework Equations The Attempt at a Solution