What is Cauchy: Definition and 388 Discussions

Baron Augustin-Louis Cauchy (; French: [oɡystɛ̃ lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra.
A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics.

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  1. G

    Cauchy integral formula(is this enough of a proof?)

    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...
  2. M

    Cauchy Riemann conditions/equation

    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...
  3. S

    Integration techniques and Cauchy prinicpal value

    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 (...
  4. A

    Pathological PDFs. eg: ratio of normals including Cauchy.

    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...
  5. M

    Integral with Cauchy Prinicpal value

    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...
  6. J

    What is the Cauchy Product Formula in Power Series?

    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?
  7. E

    Cauchy Euler, non-homogeneous, weird condition

    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...
  8. N

    Application of the Cauchy product

    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...
  9. J

    Cauchy geometric seq. proof

    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...
  10. J

    [Complex Analysis] Help with Cauchy Integral Problem

    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...
  11. F

    What type of tensor describes strain in a cubic object?

    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...
  12. K

    Cauchy sequnce and convergence of a non-monotonic sequence.

    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...
  13. K

    Slightly Harder Cauchy Integral

    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...
  14. N

    Cauchy Integral Formula - Evaluating Singularities on a Closed Path

    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...
  15. N

    Understanding the Cauchy Integral Formula and Evaluating Complex Integrals

    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...
  16. T

    Proving a sequence is a Cauchy Sequence

    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...
  17. S

    Using Cauchy Schwartz Inequality (for Integrals)

    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...
  18. H

    (dis)prove an if and only if statement of a Cauchy sequence and and interval.

    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...
  19. I

    Why Continuous Functions Don't Preserve Cauchy Sequences

    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...
  20. A

    Evaluating Complex Integrals Using Cauchy's Integral Formula

    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...
  21. A

    MHB Show a certain sequence in Q, with p-adict metric is cauchy

    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...
  22. A

    Show a certain sequence in Q, with p-adict metric is cauchy

    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...
  23. matqkks

    Why are Cauchy sequences important in understanding limits and completeness?

    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
  24. A

    Cauchy sequence in Q not converging to zero.

    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...
  25. D

    Cauchy Method/ UC set OR Variation of Param?

    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...
  26. B

    Intro to Analysis (Boundedness of Cauchy)

    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...
  27. B

    Intro to Analysis (Cauchy)

    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...
  28. C

    Is the Sum of Two Cauchy Sequences Also Cauchy?

    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...
  29. F

    Cauchy stress principle & eigenvalues of stress tensor

    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...
  30. F

    Cauchy Integral Formula application

    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?
  31. T

    Few suggestions about cauchy inequality

    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 =...
  32. F

    Power Series Expansion of Cauchy Integral Formula on the Unit Circle

    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...
  33. D

    MHB How to Express Cauchy Integral Formula in Terms of a Power Series?

    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...
  34. S

    Understanding the Cauchy Integral Theorem for Conjugate Functions

    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
  35. G

    Proving Sum of 2 Indep. Cauchy RVs is Cauchy: A Guide

    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...
  36. Fredrik

    The limit of an almost uniformly Cauchy sequence of measurable functions

    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...
  37. G

    Proving Cauchy Density Function for Z = X+Y

    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 =...
  38. alexmahone

    MHB Proof of Cauchy Sequence for $\{a_n\}$ Defined by $f(x)$

    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 $f(x)=e^{-x}$.
  39. T

    Proving the Sequence of Real Numbers is Not Cauchy

    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) <...
  40. S

    Proving Cauchy Sequence with Triangle Inequality

    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...
  41. A

    Subsequence of a cauchy sequence in R

    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...
  42. M

    Convergence and Continuity of Cauchy Sequences with Fixed Points

    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
  43. M

    Prove cauchy sequence and thus convergence

    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.
  44. R

    Cauchy sequences and continuity versus uniform continuity

    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...
  45. C

    Limits, geometric series, cauchy, proof HELP

    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...
  46. Matterwave

    Wald's definition of the Cauchy Horizon

    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...
  47. Shackleford

    Math History: Cauchy Criterion for Sequence/Series

    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...
  48. J

    Complex Analysis, Line Integrals and Cauchy Conceptually

    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...
  49. D

    Residue Theorem, Contour Integration, and the Cauchy Principal Value

    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...
  50. M

    Cauchy problem/characteristics method with initial condition on ellipse

    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...
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