Complex analysis Definition and 756 Threads

Homework Statement Consider the equation (z-1)^23 = z^23 Show that all solutions lie on the line Re(z)=1/2 How many solutions are there Homework Equations The Attempt at a Solution Really have no idea. I figured polar form might be helpful somehow so I converted it and got...

Background: I'm a computer science major, but who has done a lot of math (real analysis, linear/abstract algebra, combinatorics, probab&stats, numerical analysis, linear programming) and currently doing undergraduate research in computational algebra/geometry. I'm taking a graduate level...

Hi, In complex analysis, is it an axiom that iy=yi where y is real? Or can this result be proved somehow? Thank you.

Hello everyone, The question: My attempt: I'll try my hand at the analytic part if I could get some clarification on this part first. :)

Homework Statement Explain geometrically why the locus of z such that arg [ (z-a)/(z-b) ] = constant is an arc of a certain circle passing through the fixed points a and b. i tried to visualize the equation in a cartesian co-system but in doing so, i was not very successful.

I have never studied analysis as i am graduate student in engineering. Can anyone point me the elementary book on real and complex analysis preferably junior, undergraduate level book. I found this 2. Can anyone math graduate student comment or put some advice onto it. 1. Elementary Real and...

Hi! I have to understand how this integral is evaluated (it is taken from Fetter - Quantum theory of many particle systems)(14.24): http://dl.dropbox.com/u/158338/fis/fetter.pdf" in particular, i don't know how the log brach cuts are defined.. as far as I know, log branch cuts can be...

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) ?

Homework Statement I have to find a conformal map from \Omega = \{ z \in \mathbb C | -1 < \textrm{Re}(z) < 1 \} to the unit disk D(0,1) Homework Equations an analytical function f is conformal in each point where the derivative is non-vanishing specifically, we can think of linear...

Homework Statement "Show that there is no conformal map from D(0,1) to \mathbb C" and D(0,1) means the (open) unit disk Homework Equations Conformal maps preserve angles The Attempt at a Solution I don't have a clue. I thought the clou might be that D(0,1) has a boundary, and C...

Homework Statement If an analytic function vanishes on the boundary of a closed disc in its domain , show it vanishes on the full disc Homework Equations CR equations? The Attempt at a Solution Not sure how to start this one.

Hope someone could give me some help with a couple of problems. First: Proof of - A function f:G -->Complex Plane is continuous on G iff for every sequence C(going from 1 to infinity) of complex numbers in G that has a limit in G we have limit as n --> infinity f(C) = f(limit as n...

The title may be a bit vague, so I'll state what I am curious about. Since complex field is 'extension' to the real field, and in electrodynamics we use things like Stokes theorem, or Gauss theorem, that are being done on real field (differential manifolds and things like that, right?), can...

The part about Laurent series in my Complex Analysis book is somewhat vague and Wikipedia etc. didn't help me much. I am hoping someone would tell me the exact mathematical definition of a Laurent series (around a given point?) of a given function, perhaps providing an example. Also, how can...

[b]1. Let z be a complex variable. Describe the set of all z satisfying |z^2-z|<1.[\b] I have a `brute force' solution, but it's really messy. Without a graphing utility, it would be nearly impossible to graph. I just computed |z^2-z| in terms of x and y, and solved |z^2-z|=1 in this...

hi!. I have been looking for good complex analysis text. But, unfortunately, I haven't found it yet. Could you recommend some complex analysis textbooks except those books whose authers are silverman, alfors, churchill, conway ??

I've been trying to work through this and see whether you can take an "area" in the complex plane, have x,y vary in some interval, and integrate complex functions over that "area." The math doesn't seem to work out; plus intuitively, if you're going to sum up a function in a complex variable...

Hi all, I am currently a 2nd year mathematics and physics student. I am working, for the first time, on my own research and just sort of getting my feet wet. I got in touch with a professor that studies Special Functions and he led me to the Legendre functions and associated Legendre...

given the function arg\xi(1/2+is) is this an increasing function of 's' ?? , i mean if its derivative is always bigger than 0 here xi is the Riemann Xi function http://en.wikipedia.org/wiki/Riemann_Xi_function could we define the 'inverse' (at least for positive s) of...

Homework Statement i) Find a suitable formula for log z when z lies in the half-plane K that lies above the x-axis, and from that show log is holomorphic on K ii) Find a suitable formula for log z when z lies in the half-plane L that lies below the x-axis, and from that show log is...

I'm a bit lost on this part of my course (ODE's and complex analysis). We've only done about 2-3 of these (seemingly simple) problems where we're given the equation of a line or circle in the complex plane and are asked to find its image in the U-V plane with some transformation \omega, but I...

Hello, I am wondering what I should brush up on for a class in Complex analysis and Diff Equations. I am planning to take these in the fall and this will be by far the toughest math I will have had. I took a 4 credit Calc II with a solid A. Currently taking Calc III (through Green, Stokes and...

Homework Statement solve integral x^3/(e^x-1) with limits from 0 to infinity Homework Equations The Attempt at a Solution i tried using a rectangular contour,the boundaries of the contour pass through z=0 but the complex equivalent has pole at z=0. by Cauchy theorem the function...

Homework Statement Let f be analytic throught C, suppose that |f(z)|<=M|z|^n for a real constant M and positive integer n. Show that f is a polynomial function of degree less than n.

Homework Statement Let f:C\rightarrowC be differentiable, with f(z)\neq0 for all z in C. Suppose limf(z) is exist and nonzero as z tends to z0. Prove that f is constant.

[b]1. find the number of solutions of e^iz - z^2n - a = 0 in the upper half of the complex plane, where n is a natural number and a is a real number such that a>1. [b]2. Rouche's theorem: If f and g are analytic functions in a domain, and |f|>|g| on the boundary of the domain, then the...

S is a star-shaped open subset of \mathbb{C}, f is a holomorphic function from S to \mathbb{C}, z_0 is an element of S. I've just come out an exam and wondered whether the following 2 statements are true or false: 1 Let g be a holomorphic function on S \subseteq \mathbb{C}, with the...

http://www2.imperial.ac.uk/~bin06/M2...nation2008.pdf Solutions are here. http://www2.imperial.ac.uk/~bin06/M2...insoln2008.pdf My first question is about 3(ii), the proof of Cauchy's integral formula for the first derivative. The proof here uses the deformation lemma (from second...

In a lecture today we evaluated a integral: \oint_{\Gamma} \dfrac{3z - 2}{z^2 - z} dz Where, \Gamma = \{ z \in \mathbb{C} | |z| + |z-1| = 3 \} Our lecturer evaluated it to be 6πi I sort of understood how he did it, but he really rushed through his steps.

Homework Statement sketch the curve in the z-plane and sketch its image under w=z^2 |z-1|=1 Homework Equations z=|z|e^(iArgz) argw=2argz The Attempt at a Solution At first I simply sketched the solution for a circle centered at (1,0) in the z-plane and then mapped that to...

You can choose to limit yourself to continuous or analytical functions

Homework Statement Evaluate the integral with respect to x from 0 to infinity when the integrand is x^2/(1+x^6), using complex integration techniques. Homework Equations The Attempt at a Solution I have no idea where to start. Please help!

Hello! I know that the theory of complex analysis is useful to compute integrals of real valued functions. I am a Physics student and I followed a Complex Analysis course but we did not have time to cover this up. I am looking for a textbook that takes a practical approach to this subject. I...

Homework Statement The question asks me to find the integral from 0 to infinity of 1/(x^3 + 1), where I have to use the specific contours that they specify. Now I know that I need to use residues (in fact just one here) and the singular point is (1+sqrt(3)*i)/2. Once I can factor the (x^3...

Homework Statement |a| < 1 a is arbitrary, then show that |z| \leq 1 iff \frac{z-a}{1-a(bar)z} \leq 1 Homework Equations possible the triangle inequality The Attempt at a Solution \frac{z-a}{1-a(bar)z} is analytic everywhere except at 1/a(bar) |z - a|2 \leq |1-a(bar)z|2...

As the title says, I was wondering what would be a good book in Complex Analysis at the Undergraduate Level? I have one or two of them but like neither of them.

Homework Statement Find the Maclaurin series representation of: f(z) = {sinh(z)/z for z =/= 0 } {0 for z = 0 } Note: wherever it says 'sum', I am noting the sum from n=0 to infinity. The Attempt at a Solution sinh(z) = sum [z^(2n+1)/(2n+1)!]...

hey there there is this thing we learn in complex analysis (and almost everywhere) that if a function is analytic in a known region, then the integral on a closed path(say, any loop), will be zero. so there is another statement we need to deal with hear, which is exactly the opposite, that if...

Homework Statement If f,g are entire functions and |f(z)| <= |g(z)| for all z, draw some conclusions about the relationship between f and g Homework Equations none The Attempt at a Solution I just need a push in the right direction.. thanks for any and all help!

Homework Statement Compare the function f(z) = (pi/sin(pi*z))^2 to the summation of g(z) = 1/(z-n)^2 for n ranging from negative infinity to infinity. Show that their difference is 1) pole-free, i.e. analytic 2) of period 1 3) bounded in the strip 0 < x < 1 Conclude that they are...

Homework Statement Let f and g be two holomorphic functions in the unit disc D1 = {z : |z| < 1}, continuous in D1, which do not vanish for any value of z in the closure of D1. Assume that |f(z)| = |g(z)| for every z in the boundary of D1 and moreover f(1) = g(1). Prove that f and g are the...

Homework Statement Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations Cauchy inequalities (estimates from the Cauchy integral formula)The...

Homework Statement Let f be a holomorphic function in the unit disc D1 whose real part is constant. Prove that the imaginary part is also constant. Homework Equations Cauchy Riemann equations The Attempt at a Solution Hi guys, I'm working through the basics again. I think here we...

Homework Statement We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal...

Homework Statement Suppose f(z) is entire and the harmonic function u(x,y) = Re[f(z)] has an upper bound u_0. (i.e. u(x,y) <= u_0 for all real numbers x and y). Show that u(x,y) must be constant throughout the plane. The Attempt at a Solution Since f(z) = u(x,y) + iv(x,y) is entire...

What undergraduate math classes would you want to take if you wanted to be exposed to the stuff used in complex analysis? (Besides complex analysis)

I have to integrate S cos^8 (t) dt from 0 to 2 pi, presumably using complex analysis I got to S [(e^(it) + e(-it))/2]^8 dt from 0 ti 2pi How do I take it from here? I have a hint- use binomial theorem.

Homework Statement I'm not very good with LaTeX and the reference button seems to broken. So Assume lim h(z) = 1+i, as z->w, prove there exists a delta, d>0 s.t. 0<|z-w|<d -> (2^.5)/2 < |h(z)| < 3(2^.5)/2 Homework Equations The Attempt at a Solution Kinda been running in...

Its question 1(g) in the picture. My work is shown there as well. This has to do with independence of path of a contour. Reason I am suspicious is that first there is a different answer online and second it says "principal branch" which I have not understood. Does that mean a straight line for...

I was hoping someone could clarify the idea of a branch cut for me. In class, my professor talked about how a branch cut is used to remove discontinuities. He gave an example of |z|=1 needing a branch cut along the positive real axis. If this because going from 0 to 2\pi, the 0 and 2\pi match up?