Complex analysis Definition and 784 Threads

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?

Homework Statement Demonstrate that a domain D\in\mathbb{C} is simply connected if and only if, for every function f which is analytic and free of zeroes in D, a branch of the square root of f exists in D. Homework Equations The Attempt at a Solution I know that by definition...

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?

Hey guys, i just started a complex analysis course this semester and we just went over CR-equations, and various ways to show that a function is holomorphic. I'm a bit stuck on this one homework question where we have to prove the function is entire. Homework Statement so we have...

Hi all, I would like some recommendations for thermodynamics. It's my first course in thermo. I'm currently using : Classical and Statistical Thermodynamics by Ashley Carter. I like the book, however it lacks examples! I am someone who learns by example...so this book isn't doing me much...

this question doesn't seem tough but i can't find anything like it. \int\frac{e^{ax}}{1+e^{x}}dx along the real line (a is between 1 and 0). I know this is a complex analysis question, so i took the complex integral (along a semicircle where the diameter is the real numbers). by residue...

The problem Find Res(f,z1) With: f(z)=\frac{z}{(z^2+2aiz-1)^2} The attempt at a solution The singularities are at A=i(-a+\sqrt{a^2-1}) and at B=i(-a-\sqrt{a^2-1}) With the normal equation (take limit z->A of \frac{d}{dz}((z-A)^2 f(z)) for finding the residue of a pole of order 2, my attempt...

I am trying to decipher what this means: F(z) = \overline{f(\bar{z})} Thanks for the help.

Hello all, I'm curious as to the opinion of some people here about what is more important: Complex analysis or EM II for someone interested in going into theoretical physics (mainly particle theory). I have a hectic workload for next semester. I'm taking particle physics, EM II, grad...

Homework Statement Given that z_{1}z_{2} ≠ 0, use the polar form to prove that Re(z_{1}\bar{z}_{2}) = norm (z_{1}) * norm (z_{2}) \Leftrightarrow θ_{1} - θ_{2} = 2n∏, where n is an integer, θ_{1} = arg(z_{1}), and θ_{2} = arg(z_{2}). Also, \bar{z}_{2} is the conjugate of z_{2}. Homework...

Homework Statement Compute the contour integral I around the following curve $\Gamma$: $ I = \int_\Gamma \dfraq{dz}{z^2 +1} $ see picture: http://dl.dropbox.com/u/26643017/Screen%20Shot%202012-01-07%20at%2010.39.58.png Homework EquationsThe Attempt at a Solution $\Gamma$ is an open curve...

There is a proof offered in the text "Theory of Functions of a Complex Variable" by Markushevich that I have a question about. Some of the definitions are a bit esoteric since it is an older book. Here "domain" is an open connected set (in \mathbb{C}, in this case.) The proof that...

I am working on a problem to evaluate integrals with simple poles offset by ε above/below the real axis. So something like this ∫ [ f(x) / (x-x0-iε) ] The answer is the sum of two integrals: the principal value of the integral with ε=0 plus the integral of iπδ(x-x0). I have done the...

Homework Statement Find the residue at each pole of zsin(pi*z)/(4z^2 - 1)Homework Equations An isolated singular point z0 of f is a pole of order m if and only if f(z) can be written in the form: f(z) = phi(z)/(z-z0)^m where phi(z) is analytic and nonzero at z0. Moreover, Res(z=z0) f(z) =...

[Complex Analysis] Conformal mapping of a column into a line Homework Statement I'm having a problem with this problem (:smile:) where I have to transform an area of 2 circles, the one contained within the other, into a straight line. I've managed to transform the circles using a Möbius...

Hello, Differentiability of f : \mathbb C \to \mathbb C is characterized as \frac{\partial f}{\partial z^*} = 0. More exactly: \frac{\partial f(z,z^*)}{\partial z^*} := \frac{\partial f(z[x(z,z^*),y(z,z^*)])}{\partial z^*} = 0 where z(x,y) = x+iy and x(z,z^*) = \frac{z+z^*}{2} and...

When discussing the i (the imaginary unit) in a math class, my math teacher commented that that complex analysis is used in studying electrical circuits. I know a little about resistors and what not, but never have I seen complex analysis used this way. I've tried looking it up, but it's been...

Let D ⊂ ℂ be a domain and let f be analytic on D. Show that if there is an a ∈ D such that the kth derivative of f at a is zero for k=n, n+1, n+2,..., then f is a polynomial with degree at most n. So I believe I have a proof, but the theorems are so powerful I feel like I might be...

Homework Statement If f(z) is an entire function such that f(z)/z is bounded for |z|>R, then f''(z_0) = 0 for all z_0. Homework Equations Liouville's theorem Cauchy estimates: Suppose f is analytic for |z-z_0| ≤ ρ. If |f(z)|≤ M for |z-z_0| = ρ then the mth derivative of f at z_0 is...

Im trying to take the integral, using substitution, of \int_0^1\frac{2\pi i[cos(2\pi t) + isin(2\pi t)]dt}{cos(2\pi t)+isin(2\pi t)} So I set u=cos(2\pi t)+isin(2\pi t)du=2\pi i[cos(2\pi t) + isin(2\pi t)]dt Yet when I change the endpoints of the integral I get from 1 to 1, which doesn't make...

Homework Statement compute I=∫_2^∞ (1/(x(x-2)^.5)) dx using the calculus of residues. be sure to choose an appropriate contour and to explain what happens on each part of that contour. Homework Equations transform to a complex integral I= ∮ (1/(z(z-2)^.5)) dz The Attempt...

Homework Statement Compute the integral from 0 to 2∏ of: sin(i*ln(2e^(iθ)))*ie^(iθ)/(8e^(3iθ)-1) dθ (Sorry for the mess, I don't know how to use latex) Homework Equations dθ=dz/iz sinθ = (z - z^(-1))/2i The Attempt at a Solution So I tried to change it into a contour integral of a...

Q:Let f be entire and suppose that I am f(z) ≥ M for all z. Prove that f must be a constant function. A: i suppose M is a constant. So I am f(z) is a constant which means the function is a constant. Am i doing this right ? but i don't think there will be such a stupid question in my...

compute the integral ∫Cr (z - z0)n dz, with an integer and Cr the circle │z - z0│= r traversed once in the counterclockwise direction Solution: A suitable parametrization for Cr is give by z(t)= z0 + reit 0≤t≤2π ... ... My question is , how to find that suitable z(t)? i have no idea...

Homework Statement Let f be a quadratic polynomial function of z with two different roots z_1 and z_2. Given that a branch z of the square root of f exists in a domain D, demonstrate that neither z_1 nor z_2 can belong to D. If f had a double root, would the analogous statement be true?Homework...

Homework Statement Use partial fractions to rewrite: (2z)/(z^2+3) Homework Equations noneThe Attempt at a Solution I did this: (2z)/(z^2+3) = (Az+B)/(z^2+3) 2z = Az +B A = 2, B = 0...problem is that it just recreates the original Here is their example in the book: 1/(z^2+1) =...

Let n ≥ 2 be a natural number. Show there is no continuous function q_n : ℂ → ℂ such that (q_n(z))^n = z for all z ∈ ℂ. The only value of this function we can deduce is q_n(0)=0. Moreover any branch cut we take in our complex plane will touch zero. These two facts would make me a bit...

1) How do you integrate 1/ [z^2] over the unit circle? After you integrate, do you put it in polar form or do you replace z with x + iy then solve it? I keep getting zero. It should exist since z=o is undefined, right? 2) How do you integrate x dz over gamma, when gamma is the...

In Mathematics of Classical and Quantum Mechanics by Byron and Fuller, they state that "Some authors (never mathematicians) define an analytic function as a differentiable function with a continuous derivative." ..."But this is a mathematical fraud of cosmic proportions.. " Their main point...

I am just trying to get the conceptual basics in my head. Can I think of things this way... If you are taking the integral of a function f(z) along a curve γ in a region A. If the curve is closed and f(z) is analytic on the entire curve as well as everywhere inside the curve, then the...

1.Evaluate ∫C Im(z − i)dz, where C is the contour consisting of the circular arc along |z| = 1 from z = 1 to z = i and the line segment from z = i to z = −1. 2. Suppose that C is the circle |z| = 4 traversed once. Show that §C (ez/(z+1)) dz ≤ 8∏e4/3 For question 1, should i let z=...

Hi All, I am trying to learn complex analysis on my own and for this I have chosen Fundamentals of Complex Analysis by Saff and Snider. I am stuck at the last question in section 1.3 which is as follows. For the linkage illustrated in the figure, use complex variables to outline a scheme...

Hi All, I am trying to learn complex analysis on my own and for this I have chosen Fundamentals of Complex Analysis by Saff and Snider. I am stuck at the last question in section 1.3 which is as follows. For the linkage illustrated in the figure, use complex variables to outline a scheme...

I have a homework question: Find (1 − \sqrt{3} i)1−i and P((1 − \sqrt{3} i)1−i). I don't know what does the P stand for, And i can't find it in the textbook either. Thanks

[b]1. Let f(z) = (3e2z−ie-z)/(z2−1+i) . Find f′(1 + i). 3. Should I sub (1+i) to z and then diff it by i. Or i need to diff it by z first then sub (1+i) in it at last? Thanks

Homework Statement Let f=u+iv be an entire function. Prove that if u is bounded, then f is constant. Homework Equations Liouville's Theorem states that the only bounded entire functions are the constant functions on \mathbb{C} The Attempt at a Solution I know that if u is bounded...

One last simple question about complex analysis... Hi, sorry again for having made so many threads. I have one remaining question about complex analysis that I keep get confused on. Say that I have some complex function h(z). Sometimes I am really confused how to break that down into...

Homework Statement Considering the appropriate complex integral along a semi-circular contour on the upper half plan of z, show that \int^{\infty}_{\infty} \frac{cos(ax)}{x^2 + b^2} dx = \frac{\pi}{b}e^{-ab} (a>0, b>0) Homework Equations \int_{C} = 0 For C is a semi-circle of...

I don't get the meaning of "connected" in the chapter of planar sets. The textbook said " An open set S is said to be connected if every pair of points z1, z2 in S can be joined by a polygonal path that lies entirely in S" So do i just randomly pick 2 points in S to check if they are both in...

I'm wondering whether I could take graduate level complex analysis this spring. I planned on taking complex variables (undergraduate course), but unfortunately it conflicts with another course I want to take. I'm currently taking basic real analysis (not at the level of Rudin), point-set...

Homework Statement Demonstrate that \int_{-\gamma} f(z)|dz|=\int_{\gamma} f(z)|dz| where \gamma is a piecewise smooth path and f is a function that is continuous on |\gamma|. Homework Equations The Attempt at a Solution This proof seems like it should be very simple, but I am...

Homework Statement I have a problem as follows: Let \gamma=\beta+[e^2\pi,1] where \beta is given by \beta(t)=e^{t+it} for 0\leq 2 \leq \pi. Evaluate \int_\gamma z^{-1} dz . Homework Equations The Attempt at a Solution I know that I need to parameterize the path and I have...

Hey, I've been going through a few past papers for an upcoming exam on complex analysis, I found this T/F question with a few parts I'm not confident on, I'll explain the whole lot of my work and show.[PLAIN]http://img404.imageshack.us/img404/2069/asdasdsu.jpg a) |2+3i|=|2-3i| so false b)...

i am trying to solve below problem but not getting start; so please help The function f(z) = e^(z+i*pi) has infinitely many points in the fiber of each point in its range. (A) Find four points that map to 1 (B) The natural inverse of f(z), say g(z) maps each point in its domain to infinitely...

I don't even know where to start or go with this first problem. A) Assume that 'a sub n' E C and consider rearrangements of the convergent series the 'sum of 'a sub n' from n=1 to infinity'. Show that each of the following situations is possible and that this list includes all possibilities...

Homework Statement Find the maximum of \left|f\right| on the disc of radius 1 in the Complex Plane, for f(z)=3-\left|z\right|^{2} Homework Equations The maximum modulus principle? The Attempt at a Solution Since |z| is a real number, then surely the maximum must be 3 when z=0...