Cauchy Definition and 388 Threads

Homework Statement Prove that if f(z) is analytic over a simply connected domain containing a simple closed curve C abd Z_{0} is a point inside C then f'(z_{0}) = \frac{1}{2i\pi} \oint_{c} \frac{f(z)}{(z-z_0)^2} dz Homework Equations The Attempt at a Solution from the definition...

In the proof of the the Cauchy-Riemann's conditions we have and equality between differentials of the same function (f(z)) by x(real part) and by iy(imaginary part?). Why do we "say" that both differentials should be equal when it's normally possible to have different differentials according...

Is there a good reference that summarizes what common integration techniques (e.g. change of variables, integration by parts, interchange of the order of integration) can be used on integrands when one is calculating the Cauchy principal value (...

Hi all, I've been having a discussion about doing calculations on data which is supposedly Gaussian. And (Of course) there is a problem: Once operations are performed on the measurements -- such as taking a ratio of one kind of measurement to another; the result is often no longer a Gaussian; In...

Hi everyone, I would like to calculate the following integral: P\int_0^{\pi}\frac{1}{cos(x)-a}dx, with |a|\leq 1. The 'P' in front stands for the so-called Cauchy Principle value. Whenever a is not in the specified domain, the integrand does not have a pole and one can do the integration...

Rudin motivates this formula by multiplying two power series and then setting z = 1 and somehow obtaining the cauchy product formula. But I am not following how he does this at all. Can anyone help me understand this?

xy''+y'=-x y(1)=0, y(0) bounded (so the natural log, 1/x etc. terms drop out) homogeneous, cauchy euler: y=a+bx variation of parameters, and using the conditions gives y=1-x, I think (i tried this previously and I think this is what I got, I didnt write it down). Very different from what I...

Homework Statement Hello, I'm trying to find the Taylor representation of a product of functions - the exponential of x times sin(y). Also for (x - y)sin(x+y). The Attempt at a Solution Well, I want to use the Cauchy product in both cases. I know the taylor representation of both...

Homework Statement Suppose the sequence (Sn) is defined as: |Sn+1-Sn|<2-n show that this is a cauchy sequence Homework Equations hint: prove the polygon identity such that d(Sn,Sm)≤d(Sn,Sn+1)+d(Sn+1,Sn+2)...+d(Sm-1,Sm) The Attempt at a Solution I have defined Sm and Sn and created the...

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...

dear all, as a newbie in solid mechanics modelling, i always come across these few terms, Cauchy-Green strain tensor Green Lagrange strain tensor isochoric Cauchy green strain tensor. Consider a cubic, when we move the top face, while fixing the bottom face, we will able to see the...

Homework Statement Hello, I have a question concerning convergence of the non-monotonic sequences which takes place when the Cauchy criterion is satisfied. I understand that |a_n - a_m| <ε for all n,mN\ni Homework Equations What I don't see is how (a_{n+1} - a_n) →0is not...

Homework Statement Evaluate the integral I_1 = \int_0^{2\pi} \frac{d\theta}{(5-3sin\theta)^2} Homework Equations The Attempt at a Solution I start off by switching the sine term for a complex exponential e^{i\theta}=cos\theta +isin\theta I will consider only the Imaginary...

Cauchy Integral Formula - Multiple Repeated Poles Homework Statement C.I.F is doing my head in. Evaluate ∫ {\frac {{z}^{2}+1}{ \left( z-3 \right) \left( {z}^{2}-1 \right) }} For the closed path |z| = 2 The Attempt at a Solution This is a circle of radius 2, with...

Homework Statement Evaluating using CIF. |z| = 4 Integral {\frac {{{\rm e}^{2\,iz}}{\it dz}}{ \left( 3\,z-1 \right) ^{2}}} The Attempt at a Solution So the singularity here is z = 1/3 which is inside the circle. Therefore using the formula 2\,i\pi \,f and substituting in the...

Homework Statement Prove \sum\frac{(-1)^k}{k^2} is a Cauchy sequence. Homework Equations Definition of Cauchy sequence: |a_{n} - a_{m}|<ε for all n,m>=N, n>m The Attempt at a Solution I thought if I could prove that the above summation was less than the summation of 1/k^2, the...

Homework Statement Suppose \int_{-\infty}^{\infty}t|f(t)|dt < K Using Cauchy-Schwartz Inequality, show that \int_{a}^{b} \leq K^{2}(log(b)-log(a)) Homework Equations Cauchy Schwartz: |(a,b)| \leq ||a|| \cdot ||b|| The Attempt at a Solution Taking CS on L^{2} gives us...

Homework Statement A sequence (Xn) is Cauchy if and only if, for every ε>0, there exists an open interval length ε that contains all except for finitely many terms of (Xn). Homework Equations The Cauchy Definition is: A sequence X = (xn) of real numbers is said to be a Cauchy sequence...

Homework Statement Why is it that continuous functions do not necessarily preserve cauchy sequences. Homework Equations Epsilon delta definition of continuity Sequential Characterisation of continuity The Attempt at a Solution I can't see why the proof that uniformly continuous...

Homework Statement Use Cauchy's integral formula to evaluate when a) C is the unit circle b) c is the circle mod(Z)=2 Homework Equations I know the integral formula is The Attempt at a Solution for the unit circle I was attempting F(z)=sin(z) and Z0=∏/2, which would give a...

I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...

I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...

Why are Cauchy sequences important? Is there only purpose to test convergence of sequences or do they have other applications? Is there anything tangible about Cauchy sequences

I have the following exercise: Let s_n be a cauchy sequence in Q(rationals) not converging to 0. Show that there exists an e(epsilon) >0 and a natural number N such that either for all n>N, s_n > e or for all n>N, -s_n >e. I know that since Q is not complete, we cannot assume that there...

Hello new to this forum , Was solving some Diff eq problems and iam getting two different answers using two methods, ok the problem is i=primes (x^2)(y^ii)+(x)(y^i)+y=4sin(lnx) This is cauchy method, When i use variation of parameters i get a long answer with impossible integrals and when i...

Homework Statement Prove the Boundedness Theorem for Cauchy Sequences by a contrapositive argument. Homework Equations The Attempt at a Solution If {an} is not bounded then {an} is not a cauchy sequence. We will prove that {an} is not a cauchy sequence by showing that there...

Homework Statement Prove the following assertion: Suppose {xn} and {yn} are Cauchy sequences of real number. If {xn} is a cauchy sequence and for every η>0 there exists a pos. int. N such that for every n>N so that abs(xn-yn)<η then {yn} is a Cauchy sequence. Homework Equations None...

Homework Statement Assume x_n and y_n are Cauchy sequences. Give a direct argument that x_n+y_n is Cauchy. That does not use the Cauchy criterion or the algebraic limit theorem. A sequence is Cauchy if for every \epsilon>0 there exists an N\in \mathbb{N} such that whenever...

First of all, thanks for all the helpful comments to my previous posts. I'm trying to get a grasp of stress tensors and have been doing some studying. In the literature I've been looking at, it says something about the eigenvalues of stress tensors and the principle stresses. This is...

f is analytic on an open set U, z_0\in U, and f'(z_0)\neq 0. Show that \frac{2\pi i}{f'(z_0)}=\int_C\frac{1}{f(z)-f(z_0)}dz where $C$ is some circle center at $z_0$. S0 ,f(z)-f(z_0) = a_1(z-z_0)+a_2(z-z_0)^2+\cdots with a_1=f'(z_0)\neq 0. But why can f(z)-f(z_0) be expanded this way?

As I can see from the formula of cauchy inequality: (a1^2+a2^2+...+an^2)^1/2 . (b1^2+b2^2+...+bn)^1/2 >= a1b1+a2b2 + ... + anbn Can I conclude from the above formula that: (a1+a2+...+an)^1/2 . (b1+b2+...+bn)^1/2 >= (a1b1)^1/2 + (a2b2)^1/2 +...+ (anbn)^1/2 by setting a1,...,an =...

For all z inside of C (C the unit circle oriented counterclockwise), f(z) = \frac{1}{2\pi i}\int_C \frac{g(u)}{u-z} du where g(u) = \bar{u} is a continuous function and f is analytic in C. Describe fin C in terms of a power series. \displaystyle f(z) = \frac{1}{2\pi i}\int_C...

For all z inside of C (C the unit circle oriented counterclockwise), $$ f(z) = \frac{1}{2\pi i}\int_C \frac{g(u)}{u-z} du $$ where $g(u) = u^7$ is a continuous function and $f$ is analytic in C. Describe $f$ in C in terms of a power series. $\displaystyle f(z) = \frac{1}{2\pi i}\int_C...

Homework Statement ∮ dz/(2 - z*) over Curve |z|=1? where z* = conjugate of z How to solve this? Homework Equations I tried doing by taking z*=e^(-iθ) , the answer was zero Then i did it by taking z*=1/z which gives ∏i/2. The Attempt at a Solution

Given the fact that X and Y are independent Cauchy random variables, I want to show that Z = X+Y is also a Cauchy random variable. I am given that X and Y are independent and identically distributed (both Cauchy), with density function f(x) = 1/(∏(1+x2)) . I also use the fact the...

The limit of an "almost uniformly Cauchy" sequence of measurable functions I'm trying to understand the proof of theorem 2.4.3 in Friedman. I don't understand why f must be measurable. The "first part" of the corollary he's referring to says nothing more than that a pointwise limit of a...

Homework Statement Let X and Y be independent random variables each having the Cauchy density function f(x)=1/(∏(1+x2)), and let Z = X+Y. Show that Z also has a Cauchy density function. Homework Equations Density function for X and Y is f(x)=1/(∏(1+x2)) . Convolution integral =...

Suppose $f(x)$ is continuous and decreasing on $[0, \infty]$, and $f(n)\to 0$. Define $\{a_n\}$ by $a_n=f(0)+f(1)+\ldots+f(n-1)-\int_0^n f(x)dx$ (a) Prove $\{a_n\}$ is a Cauchy sequence directly from the definition. (b) Evaluate $\lim a_n$ if $f(x)=e^{-x}$.

Homework Statement Show that the sequence of real numbers defined by x_{n + 1} = x_n + \frac{1}{x_n^2}, \, x_1 = 1 is not a Cauchy sequence. Homework Equations A sequence \{ p_n \} is Cauchy if and only if, for all \varepsilon > 0, there exists an N > 0 such that d(p_n, p_m) <...

Homework Statement If a sequence {xn} in ℝn satisfies that sum || xn - xn+1 || for n ≥ 1 is less than infinity, then show that the sequence is Cauchy. Homework Equations The triangle inequality? The Attempt at a Solution || xm - xn || ≤ || Ʃ (xi+1 - xi) from i=n to m-1|| Using...

Homework Statement If \{a_{n}\}\in\mathbb{R} is Cauchy, \forall\epsilon>0,\exists a subsequence \{a_{k_{j}}\} so that |a_{k_{j}}-a_{k_{j+1}}|<\frac{\epsilon}{2^{j+1}}. The Attempt at a Solution Since \{a_{k_{j}}\} is Cauchy,\forall\epsilon>0,\exists N_{\epsilon} such that for j,j+1\geq...

Let f : [a,b] → [a,b] satisfy |f(x)-f(y)| ≤ λ|x-y| where 0<λ<1. Prove f is continuous. Choose any Xo ε [a,b] and for n ≥ 1 define X_n+1 = f(Xn). Prove that the sequence (Xn) is convergent and that its limit L is a 'fixed point' of f, namely f(L)=L

Let (Xn) be a sequence satisfying |Xn+1-Xn| ≤ λ^n r Where r>0 and λ lies between (0,1). Prove that (Xn) is a Cauchy sequence and so is convergent.

Homework Statement This isn't really a problem but it is just something I am curious about, I found a theorem stating that you have two metric spaces and f:X --> Y is uniform continuous and (xn) is a cauchy sequence in X then f(xn) is a cauchy sequence in Y. Homework Equations This...

i guys, I'm stuck on wording of a homework assignment and thought you might be able to help me. There are several questions... Consider the geometric series: (Sum from k=0 to infinity) of ar^k and consider the repeating decimal .717171717171 for these problems: Question 1: Find a formula...

Hello, Wald defines, on page 203 the future Cauchy Horizon of a set S\subset M as: H^+(S)=\overline{D^+(S)}-I^-[D^+(S)] Where the overline means the closure of the set. D+ is the future domain of dependence (i.e. all points in the manifold which can be connected to S by a past inextendible...

I know the Cauchy criterion for a convergent sequence. A Cauchy sequence is one in which the distance between successive terms becomes smaller and smaller. You can find a number N such that the terms after that, pairwise, have a a distance that is less than epsilon. After looking at an...

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...

Hi Folks, I worked out a couple of problems on finding the Cauchy Principal Value, and I would like to check whether my solutions are correct and also take the opportunity to ask a couple of general questions about the residue theorem, contour integration, and the Cauchy principal value. The...

Homework Statement Consider the PDE xu_x + y u_y = 4 u, -\infty < x < \infty, -\infty < y < \infty. Find an explicit solution that satisfies u = 1 on the ellipse 4x^2 + y^2 = 1. Homework Equations The Attempt at a Solution The characteristic curves are x(t,s) = f_1(s) e^t...