Complex analysis Definition and 755 Threads

Hey everyone, I'm a math major and I am having a lot of trouble in my Complex Analysis class right now. I studied for 3 hours today for our first quiz on topology of the complex plane and I got somewhere between 40-50%. I memorized all the definitions but screwed up drawing the regions...

Homework Statement 1)Show that (1-i)^{2}=-2i then evaluate (1-i)^{2004}+(1-i)^{2005} 2)Prove that every complex number with moduli 1, except z=1, can be put in the form \frac{a+i}{a-i} 3)Let m and n be positive integers without a common factor. Define z^{m/n}=(z^{m})^{1/n}, and show...

Having just gone through a section of complex analysis in a math course, I'm curious when you would actually use things like contour integration and residue theory in EE. I've been told complex analysis has all these applications in z and laplace transforms, but it seems like you only ever...

Hi everyone, I'm a Physics student going into my Junior year and I'm currently registering for my courses for the following semester and I have two options for my "complex" course, namely: --------------------------------------------------- Complex Variables Theory of functions of one complex...

First post. Great to be here. :) So, I'm stuck in deciding which of these courses to take next semester. I'm a current rising sophomore at UT Austin who has just switched from EE to physics-still nervous about that decision, but that's a separate topic. I've already got Waves(the first...

Homework Statement http://img684.imageshack.us/img684/779/334sn.jpg The Attempt at a Solution The first part was fairly straightforward, solve for z + 1, and then get w in terms of u + iv, rationalise the denominator, and then we get (x,y) in terms of u and v, which we substitute back...

Homework Statement I have the following problem: Compute \operatorname{Re} \int _\gamma \frac{\sqrt{z}}{z+1} dz, where \gamma is the quarter-circle \{ z: |z|=1, \operatorname{Re}z \geq 0 , \operatorname{Im} z \geq 0 \} oriented from 1 to i, and \sqrt{z} denotes the principal...

Homework Statement Now, I know there's two ways to go about this and it seems everywhere I look around on the web people are solving it in a way I think that seems longer, harder and more prone to mistakes in exams. It involves using the exponential identities and taking logs. I was shown...

I've been studying the residue theorem and I've been having some difficulty with classifying singularities. For example, let's use the function f(z) = \frac{1}{z sinz} I know it has two singularities, one at z=0 and the other at z=2kπ for k ={0,1,2,..}, I don't know what kind of singularities...

Homework Statement Find the radius of convergence of the Taylor series at 0 of this function f(z) = \frac{e^{z}}{2cosz-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: First, I tried to re-write it as a Laurent series to find...

Homework Statement Evaluate the following integral, I = \int_{0}^{2\pi} \frac{d \theta}{(1-2acos \theta + a^2)^2}, \ 0 < a < 1 For such, transform the integral above into a complex integral of the form ∫Rₐ(z)dz, where Rₐ(z) is a rational function of z. This will be obtained through the...

Use complex analysis to evaluate the integral [from 0 to 2∏]∫dt/(b + cost) with b < -1.

Suppose f is an entire function and, for every z in the complex plane, |f'(z) - (2 + 3i)| ≥ 0.00007. Suppose also that f(0) = 10 + 3i and f'(7+ 9i) = 1 + i. What is f(1 - 4i)?

Homework Statement Hi everyone, This is more of a definition clarification than a question. I'm just wondering if a branch is the same thing as a branch line/branch cut? I've come across a question set that is asking me to find branches, but I can only find stuff on branch lines/cuts and...

Folks, I am trying to understand calculating residues. http://www.wolframalpha.com/input/?i=residue+of+1%2F%28z%5E2%2B4%29%5E2+at+z%3D2i How is that answer determined? I mean (2i)^2=-4 and hence denominator is 0...? Thanks

If a complex function has the form: f(z) = g(z)/(z-a)n then z=a is a pole of order n. I don't really understand all this fancy terminology. Isn't a pole just like when you for a real valued function g(x)/(x-a) don't want to divide by 0 and therefore the function is defined at x=a? If so what...

I have to choose which math course I'm going to take next term. I want to take both but I'm already taking two physics courses and my college's distribution requirements require that I take an English next term... bleh... I could audit one of the physics and then take both math courses, but that...

Homework Statement Show that the function f defined by f(z) = 3\,{x}^{2}y+{y}^{3}-6\,{y}^{2}+i \left( 2\,{y}^{3}+6\,{y}^{2}+9\,x \right) is nowhere differentiable.The Attempt at a Solution Computing the C.R equations for this, I am left with {y}^{2}+2\,y={\it xy} and x^2+(y-2)^2 = 1...

Homework Statement Let F be the set of all analytic functions f that map the open unit disc D(0,1) into the set U = \left\{w=u+iv : -2 < u < 2 \right\} such that f(0)=0. Determine whether or not F is a normal family. Homework Equations DEF'N: A normal family on a domain (i.e. open and...

Homework Statement Gamelin VIII.1.6 (8.1.6) "For a fixed number a, find the number of solutions of z^5+2z^3-z^2+z=a satisfying Re z > 0" Homework Equations The argument principle relating the change in the argument to the number of zeros and poles of the function on the domain. The...

Hi all, I need to evaluate this integral anybody could point me to a solution? I've tried to look around (google, books), but I found no clue to solve it I wrote it in latex $\displaystyle \int_0^{2\pi} \! \frac{1}{(2 + \cos \theta)^2} \mathrm{d} \theta$ Thanks for the help, matteo

Homework Statement I have some past exam questions that I am confused with Homework Equations a_{n} = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz The Attempt at a Solution I'm not sure how to approach this, I'm completely lost and just attempted to solve a few: a) it says f(z)...

Homework Statement Suppose c and (1 + ic)^{5} are real, (c ≠ 0) Show that either c = ± tan 36 or c = ± tan 72The Attempt at a Solution So I considered the polar form \left( {{\rm e}^{i\theta}} \right) ^{5} and that \theta=\arctan \left( c \right) , so c = tan θ Using binomial expansion, I...

Homework Statement Suppose w is not in the interval [-R,R] show that the equation z+\frac{R^{2}}{z}=2w has one solution z with |z|<R and one solution z with |z|>R Homework Equations none The Attempt at a Solution the book mentions that the quadratic is left unchanged by the...

how to find the complex root of z^5 = 0 there is one real root 0

Hi, Everyone! I just want to ask about the importance of Complex numbers analysis in the discipline of Electronics and Communications Engineering. I'm taking a course called, Analytical Methods in Engineering, and it's mostly focused on how to deal with complex numbers, from applying algebraic...

Homework Statement Determine where the function f(x + iy) = 2sin(x) + iy^2 + 4(ix - y) is differentiable and where it is analytic.The Attempt at a Solution f(x + iy) = 2sin(x) -4y + i(y^2 +4x) Through C-R equations: du/dx = 2 cos x dv/dy = 2y du/dy = -4 dv/dx = 4 So the C-R equations hold...

$$ \int_0^{\pi}\frac{ad\theta}{a^2 + \sin^2\theta} = \int_0^{2\pi}\frac{ad\theta}{1 + 2a^2 - \cos\theta} = \frac{\pi}{\sqrt{1 + a^2}} $$ Consider $a > 0$ and $a < 0$ First I don't think the second part is correct. Shouldn't it be $1 + 2a^2 - \cos 2\theta$?

$$ \int_{0}^{\infty}\frac{\sin^2 x}{x^2}dx = \frac{\pi}{2} $$. [Hint: Consider the integral of $(1 - e^{2ix})/x^2)$.] If we look at the complex sine, we have that $\sin z = \frac{e^{iz}-e^{-iz}}{2i}$. Then $$ \sin^2z = \frac{e^{-2iz}-e^{2iz}}{4} $$ so $$ \frac{\sin^2 z}{z^2} =...

Homework Statement Let f(z) be an analytic function in the complex plane ℂ, and let \phi be amonotonic function of a real variable. Assume that U(x,y) = \phi(V(x,y)) where U(x,y) is the real part of f(z) and V(x,y) is the imaginary part of f(z). Prove that f is constant. Homework Equations...

Homework Statement If \frac{z}{z + 3} is purely imaginary, show that z lies on a certain circle and find the equation of that circle.The Attempt at a Solution So, \frac{z}{z + 3} = \frac{x + iy}{x + iy + 3} Multiplying by the complex conjugate (and simplifying), we get, \frac{x^{2} + y^{2}...

Homework Statement I just wrote a test and was wondering if I got these questions right, I already solved them, please see the attached pictures below. Here are the questions; sorry for non-latex form 1) Let gamma be a positively oriented unit circle (|z|=1) in C solve: i) integral of...

Homework Statement Simplify in terms of real and imaginary parts of x and y and sketch them. 1) Re \frac{z}{z-1} = 0 2) I am \frac{1}{z} ≥ 1 The Attempt at a Solution 1) \frac{x + iy}{x + iy -1} = 0 Am I allowed to just vanish the imaginary components here and have \frac{x}{x...

Homework Statement The point 1 + i is rotated anticlockwise through \frac{∏}{6} about the origin. Find its image. The Attempt at a Solution The point 1 + i creates an angle of arctan(1/1) = ∏/4 The rotation is by a further angle β = ∏/6. So the new point w in the w-plane from...

Homework Statement Suppose both c and (1 + ic)^{5} are real (c \neq 0). Show that c = ± \sqrt{5 ± 2\sqrt{5}} Now use another method to show that either c = ± tan 36◦ or c = ± tan 72◦ The Attempt at a Solution I expanded it out, but I'm not entirely too sure how to solve this for...

There does not exist a non-constant analytic function in the unit circle which is real valued on the unit circle. I am not able to see why. I am trying to apply Louisville's Theorem, or maybe Open Mapping Th., but I fail. Is there a way of extending this function so that it entire? and even...

Two questions: 1)Quote comes from a textbook: Each non-constant function analythic function with f(0)=0 is,in a small nbhd of 0, the composition of a conformal map with the nth-power map...The proof is given and I think I am comfortable with it.. My question is a lot simpler (I think)...

Homework Statement Given: f is an entire function, Re f(z) ≤ n for all z. Show f is constant. Homework Equations The Attempt at a Solution So I thought I'd use Liouville's Theorem which states that, if f(z) is entire and there is a constant m such that |f(z)| ≤ m for all z...

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...

Homework Statement Show that if |z| = 10 then 497 ≤ |z^{3} + 5iz^{2} − 3| ≤ 1503. The Attempt at a Solution I'm not an entirely sure how to begin this one, or if what I'm doing is correct. If I sub in |z| = 10 into the equation; |1000 + 500i - 3| = 997 +500i Then the modulus of...

Here's the problem: Let f and g be analytic functions on a simply connected domain Ω such that f2(z) + g2(z) = 1 for all z in Ω. Show that there exists an analytic function h such that f(z) = cos (h(z)) and g(z) = sin(h(z)) for all z in Ω. Here's my attempt at a solution: f2 + g2 = 1 on Ω...

I have a copy of Robert Burckel's An Introduction to Classical Complex Analysis, Volume 1. What happened to volume 2? The introduction to volume 1 contains a description of the contents of volume 2. It also contains the table of contents of volume 2. The beginning of volume 1 lists some of the...

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.

The problem is attached regards, chwala ken

Let S=[0,infinity) and let f{_n}(z)=n^2ze^-(nz) Show that f{_n} -> 0. Is the function uniformly convergent? Sorry about it being unclear but TEX tags don't see to work. f{_n} means f subscript n. Thanks

I am trying to understand a problem in my book (for reference pr 167 Serge Lang Complex Analysis). $$ f(z) = \frac{1}{z} + \sum_{n = 1}^{\infty}\frac{z}{z^2-n^2} $$ Let R>0 (is this R representing the radius of convergence?) and let N>2R (where did this come from and why?). Write $f(z) =...

This isn't really homework help. I'm working through a complex analysis textbook myself, and am stumped on the complex transcendentals, but I figured this was the best place for it. I would greatly appreciate any guidance here, I'm getting very frustrated! Homework Statement The problem is to...

Homework Statement I need to prove that \sum_{n=1}^{∞}[1−Cos(n−1z)] is entire. Homework Equations The Attempt at a Solution I know that I need to show that the series is differentiable for its whole domain, but I am not sure how to do that. Should I try to use the ratio test?

Hey all, I was reading up on Harmonic functions and how every solution to the laplace equation can be represented in the complex plane, so a mapping in the complex domain is actually a way to solve the equation for a desired boundary. This got me wondering: is this possible for other PDEs...

If we know f(x)^2 and f(x)^3 are both holomorphic, can we say that f(x) itself is also holomorphic? And how to prove that?