Cauchy Definition and 388 Threads

Hey guys, I was just doing some independent study on products of series and I'm trying to understand/derive the following form of the Cauchy product of series: \left(\sum_{n=0}^{N} a_{n}\right) \left(\sum_{m=0}^{N} b_{m}\right) = \sum_{n=0}^{N} \left(\sum_{k=0}^{n} a_{k}b_{n-k}\right)...

Show that the whole complex has zero following result (Cauchy Integral Theorem):

I have this cauchy problem U_t(x,t)= c_0[tanhx]u_x(x,t)=0 U(x,0)= u_0(x) I managed to prove that it has at most one solution my question is why would it be redundant to have a boundary condition at x=0

Hello. How do I know when to use Cauchy integral formula. Why do we use the formula in this question? As you can see in my attempt, I got stuck. What is f(z), z, z​0 here?

Using the fact that the Natural Nos are complete .then prove that every Cauchy sequence in Natural Nos converges in N and the converse. I do not even know if we can have a Cauchy sequence in Natural Nos. What would be the appropriate metric to use in our Cauchy sequence??

Hello everyone, I have recently bumped into the Kramers Kronig Relations while reviewing some of my Eletromagnetism notes, and as you may know those relations are written in terms of the Cauchy Principal Value (CPV) of certain integrals. Well, I've never been very familiar with with the...

We let C be the set of Cauchy sequences in \mathbb{Q} and define a relation \sim on C by (x_i) \sim (y_i) if and only if \lim_{n\to \infty}|x_n - y_n| = 0. Show that \sim is an equivalence relation on C. We were given a hint to use subsequences, but I don't think they are really necessary...

The cauchy principal value formula is: But why ε→0⁺ in both terms? The correct wouldn't be ε→0⁻ in 1st term and ε→0⁺ in 2nd term? Like: \lim_{\varepsilon \to 0^-}\int_{a}^{c-\varepsilon}f(x)dx + \lim_{\varepsilon \to 0^+}\int_{c+\varepsilon}^{b}f(x)dx ?

Hello, I don't get why using the fact that ∫dz/z = 2*pi*i accros the circle This integral gives: 1/3∫(1/(z-2)-1/(z-1/2))dz = 1/3(-2*pi*i) across the circle Thanks !

Homework Statement If llull = 4, llvll = 5 and u dot v = 10, find llu+vll. u and v are vectors Homework Equations llu+vll = llull + llvll cauchy schwarz The Attempt at a Solution (1) llu+vll = llull + llvll (2) (llu+vll)^2 = (llull + llvll)^2 (3) (llu+vll)^2 = llull^2 +...

Homework Statement Given: x_{n+1}=\frac{1}{3+x_n} with x_1=1 Show that: (1) |x_{n+1}-x_n| \leq \frac{1}{9}|x_{n}-x_{n-1}| and (2) x_n is Cauchy. Homework Equations The Attempt at a Solution I've tried different approaches (including induction) but the...

I know that a metric space is complete if every Cauchy sequence converges that will surely designate compact metric spaces as complete spaces . I need to see examples of metric spaces which are not complete. Thanks in advance !

Use Cauchy Integral Formula to solve: ∫ [(5z² - 3z + 2)/(z-1)³] dz C is any closed simple curve involving z=1. (z is a complex) Thanks and sorry for my poor english, it's not my first language.

Let y(t) = (y1(t), y2(t))^T and A(t) = (a(t) b(t) c(t) d(t)). A(t) is a 2x2 matrix with a,b,c,d all polynomials in t. Consider the two dimensional Cauchy problem y'(t) = A(t)y(t), y(0)=y0. Show that a solution exists for all t>=0. Give a general condition on the A(t) which ensures...

Homework Statement $$\int_\gamma \frac{\cosh z}{2 \ln 2-z} dz$$ with ##\gamma## defined as: 1. ##|z|=1## 2. ##|z|=2## I need to solve this using Cauchy integral formula. Homework Equations Cauchy Integral Formula The Attempt at a Solution With ##|z|=2## I've solved already, as it is...

one of example of cauchy sequence show that = 1/n - 1/(n+k) and In the above we have used the inequality 1/(n+m)^2 <= ( 1/(n+m-1) - 1/(n+m) ) => i don't under stand where this come from and what is inequality? can you give other example?

Hello, I am not sure what the first indice in the cauchy stress tensor indicates For example, σ_xy means that the stress in the y direction, but does x mean the cross sectional area is normal to the x direction?

A complex analysis question. Homework Statement Verify the Cauchy theorem by calculating the contour integrals. Where ω is the appropriately orientated boundary of the annulus/donut defined by 1/3 ≤ IzI ≤ 2 for the following analytic functions: i. f(z)=z^2 ii. f(z)=1/z Homework...

Homework Statement Does the Cauchy Schwarz inequality hold if we define the dot product of two vectors A,B \in V_n by \sum_{k=1}^n |a_ib_i| ? If so, prove it. Homework Equations The Cauchy-Schwarz inequality: (A\cdot B)^2 \leq (A\cdot A)(B\cdot B) . Equality holds iff one of the vectors...

The Cauchy expansion says that \text{det} \begin{bmatrix} A & x \\[0.3em] y^T & a \end{bmatrix} = a \text{det}(A) - y^T \text{adj}(A) x , where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar. There is a proof in Matrix Analysis by Horn and...

Lets say that we have two Cauchy sequences {fi} and {gi} such that the sequence {fi} converges to a limit F and the sequence {gi} converges to a limit G. Then it can easily be shown that the sequence defined by { d(fi, gi) } is also Cauchy. But is it true that this sequence, { d(fi, gi) }...

Homework Statement Let (M,d) be a complete metric space and define a sequence of non empty sets F1\supseteqF2\supseteqF3\supseteq such that diam(Fn)->0, where diam(Fn)=sup(d(x,y),x,y\inFn). Show that there \bigcapn=1∞Fn is nonempty (contains one element). Homework Equations The...

Homework Statement Show that the sequence (xn)n\inN \inZ given by xn = Ʃ from k=0 to n (7n) for all n \in N is a cauchy sequence for the 7 adic metric. Homework Equations In a metric space (X,dx) a sequence (xn)n\inN in X is a cauchy sequence if for all ε> 0 there exists some M\inN such...

Suppose (an) is sequence in the metric space X and define Tn={ak:k>n} and diamT=sup{d(a,b):a,b elements of T}. Prove that (an) is Cauchy if and only if diam Tn converges to zero. In what metric spacee does Tn converge? I assumed in (ℝ,de) but this is confusing since the diam of T is...

Alright so I posted a picture asking the exact question. Here is my best attempt... According to my professor's terrible notes, the numerator can magically turn into the form: e^i(z+3) when converted to complex. The denominator will be factored into (z-2i)(z+2i) but the...

I am looking for a different proof that ##(S_n) = \frac{1}{n}## is cauchy. The regular proof goes like this (concisely): ##\left|\frac{1}{n} - \frac{1}{m} \right| \leqslant \left|\frac{m}{nm}\right| \ (etc...) \ <\epsilon ## but I was thinking about an alternative proof. Is my proof...

I have worked my way though the proof of the Cauchy Schwarz inequality in Rudin but I am struggling to understand how one could have arrived at that proof in the first place. The essence of the proof is that this sum: ##\sum |B a_j - C b_j|^2## is shown to be equivalent to the following...

Hi everybody, I would like to use the 'cauchy dispersion formula', ie (http://en.wikipedia.org/wiki/Cauchy's_equation"]http://en.wikipedia.org/wiki/Cauchy's_equation):[/PLAIN] eta = A + B / w² Where : eta is the resulting IOR A is the base IOR B is the dispersion coefficient expressed...

I want to prove the Cauchy integral formula : \oint_{\gamma} \, \frac{f(z)}{z-z_0}\,dz= 2\pi i f(z_0) \text{so we will integrate along a circle that contains the pole .} |z-z_0|= \delta \,\text{ which is a circle centered at the pole and has a radius }\delta\,\, z =z_0+\delta e^{i\theta...

Hey, I have a problem with this integral: \int_{-\infty}^{\infty}dE\frac{1}{E^{2}-\mathbf{p}^{2}-m^{2}+i\epsilon}\: ,\: l^{2}=\mathbf{p}^{2}+m^{2} The integration over all energies (arising in the loop function for calculating the scattering), I understand we write the above in this form...

Homework Statement \int_0^\infty\frac{x^{p-1}}{1+ x}dx ** I could not get p-1 to show as the exponent; the problem is x raised to the power of p-1. \int_0^\infty\frac{ln(x) dx}{(x^2+1)^2} The Attempt at a Solution There is no attempt, but I would like to make one! I'm asking...

What does the "N" mean in a Cauchy sequence definition? Hi everyone, I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question. I believe I have an intuitive understanding of what a Cauchy...

While reading the reference Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html, there is something that I don't quite understand. Eq.(16.6) is an evolution equation for...

Homework Statement Let {pn}n\inP be a sequence such that pn is the decimal expansion of \sqrt{2} truncated after the nth decimal place. a) When we're working in the rationals is the sequence convergent and is it a Cauchy sequence? b) When we're working in the reals is the sequence...

Homework Statement Prove the following theorem, originally due to Cauchy. Suppose that (a_{n})\rightarrow a. Then the sequence (b_{n}) defined by b_{n}=\frac{(a_{1}+a_{2}+...+a_{n})}{n} is convergent and (b_{n})\rightarrow a. Homework Equations A sequence (a_{n}) has the Cauchy property...

Hello, My instructor, whilst trying to prove that liminf of sequence a_n = limsup of sequence a_n = A, _ wrote that since we know that a_n0-ε<an<a_n0+ε → a_n0-ε ≤ A ≤ A ≤ a_n0+ε...

Homework Statement evaluate the given trigonometric integral ∫1/(cos(θ)+2sin(θ)+3) dθ where the lower limit is 0 and the upper limit is 2π Homework Equations z = e^(iθ) cosθ = (z+(z)^-1)/2 sinθ = (z-(z)^-1)/2i dθ = dz/iz The Attempt at a Solution after I substitute and...

Homework Statement Compute the following integral around the path S using Cauchys integral formula for derivatives: \intez / z2 Integral path S is a basic circle around origin. Then, use the result to compute the following integral \int ecos (x) cos(sin (x) - x) dx from 0 to ∏...

Homework Statement For an quiz for a diff eq class, I need to prove the Cauchy integral formula. The assignment says prove the formula for analytic functions. Is the proof significantly different when the function is not analytic? Homework Equations Basically, a proof I found online says...

Homework Statement Let (xn)n\inℕ and (yn)n\inℕ be Cauchy sequences of real numbers. Show, without using the Cauchy Criterion, that if zn=xn+yn, then (zn)n\inℕ is a Cauchy sequence of real numbers. Homework Equations The Attempt at a Solution Here's my attempt at a proof: Let...

Homework Statement I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence Homework Equations What I know is: a[n] is bounded any subsequence is bounded there exists a monotone subsequence all monotone bounded sequences converge there exists a...

I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded I don't understand that how we can take m=N_{1} while m>N_{1} ? In fact i mean that in a metric space (A,d) can we say that...

Homework Statement Prove the converse of the Cauchy Criterion for Limits. Let I be an interval that either contains the point c or has c as one of its endpoints and suppose that f is a function that is defined on I except possibly at the point c. Then the function f has limit at c iff for...

Homework Statement Show that sin(z) satisfies the condition. (Stated in the title) Homework Equations The Attempt at a Solution f(z) = sin (z) = sin (x + iy) = sin x cosh y + i cos x sinh y thus, u(x,y)=sin x cosh y ... v(x,y)= cos x sinh y du/dx = cos x...

Hi, I came across this for the first time today. \int_0^\infty e^{i\omega t}dt = \pi\delta(\omega)+iP(\frac{1}{\omega}) Here P(\frac{1}{\omega}) is the so called principal value. I haven't seen this term normally so can I ask where we get it from? Googling principal value showed me a very...

Hello! I have a couple of questions on the following. Firstly, I was hoping someone could check my working and my reasoning. Secondly, I was wondering if someone knew an alternative way of solving this problem. I wanted to integrate this from x = -\infty to x = \infty: \lim_{a...

Homework Statement To find the integral by Cauchy Residue Theorem and apply substitution method. Homework Equations To show: ∫^{2∏}_{0}\frac{cosθ}{13+12cosθ}=-\frac{4∏}{15} The Attempt at a Solution The solution I have done is attached. It is different as what the question wants me...

Homework Statement I'm trying to follow the demonstration of the Cauchy-Schwarz's inequality proof given in http://mathworld.wolfram.com/SchwarzsInequality.html. I am stuck at the last step, namely that \langle \bar g , f \rangle \langle f , \bar g \rangle \leq \langle \bar f , f \rangle...

Homework Statement The question is needed to be done by using an appropriate substitution and the Cauchy Integral Formula. Homework Equations Evaluate the complex integral: ∫e^(e^it) dt, from 0 to 2∏ The Attempt at a Solution I cannot find an appropriate substitution for the integral.

If given a cauchy euler equation (non-homogeneous) equation, does the approach in looking for a particular solution (in order to solve the non-homogeneous part), differ from normal? I am also in general confused about how to assign a particular solution form, in many cases. I have yet to find...