Cauchy Definition and 388 Threads

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. C

    Exploring the Differences between τxy and τyx in the Cauchy Stress Tensor

    Homework Statement https://en.wikipedia.org/wiki/Cauchy_stress_tensor[/B] I don't understand the difference between τxy . τyx , τxz , τzx , τyz , τzy ..What did they mean ? Homework EquationsThe Attempt at a Solution taking τxy and τyx as example , what are the difference between them ? They...
  2. P

    Cauchy Momentum Equation Derivation

    From "Cauchy Momentum Equation" on Wikipedia, The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes Fi This is exactly what I am having trouble grasping. It's probably something simple and...
  3. K

    I Is Cauchy Reimann condition sufficient for complex differentiability?

    Hi, I have a question about Cauchy Reimann equation lets say z=x+yi is in R^2 And there exists f:R^2->R^2 f(z)=u(x,y)+v(x,y)i Then cauchy reimann condition states that If partial x of f and y of f are equal, then f is holomorphic However, I am not sure how this can be a necesary sufficient...
  4. D

    A Cauchy convolution with other distribution

    I have a set of data which are probably convolutions of a Cauchy distribution with some other distribution. I am looking for some model for this other distribution so that a tractable analytic formula results. I know that the convolution Cauchy with Cauchy is again Cauchy, but I want the other...
  5. P

    Find the residues of the following function + Cauchy Residue

    Homework Statement Find the residues of the function f(z), and compute the following contour integrals. a) the anticlockwise circle, centred at z = 0, of radius three, |z| = 3 b) the anticlockwise circle, centred at z = 0, of radius 1/2, |z| = 1/2 f(z) = 1/((z2 + 4)(z + 1)) ∫Cdz f(z) Homework...
  6. saybrook1

    Determining Cauchy principal value of divergent integrals

    Homework Statement So I've found a ton of examples that show you how to find cauchy principal values of convergent integrals because it is just equal to the value of that integral and you prove that the semi-circle contribution goes to zero. However, I need to find some Cauchy principal values...
  7. TheDemx27

    I Cauchy Repeated Integration Explanation?

    https://anhngq.wordpress.com/2013/04/25/the-cauchy-formula-for-repeated-integration/ Could someone please explain, bit by bit, how the formula works? For instance, why are we integrating with respect to sigma, up to sigma in the equation before it?
  8. ognik

    MHB Please check Cauchy Integral Formula excercise

    Find $ \oint\frac{e^{iz}}{z^3}dz $ where contour is a square, center 0, sides > 1 There is an interior pole of order 3 at z=0 CIF: $ \oint\frac{f(z)}{(z-z_0)^{n+1}}dz = \frac{2\pi i}{n!} f^{(n)}(z_0) = \frac{2 \pi i}{2}f''(z_0) = -\pi i $
  9. evinda

    MHB Solving a Cauchy Problem for $f(x,y)$, Defined on $(-b,b)$

    Hello! (Wave) We consider the following Cauchy problem $$y'=f(x,y), y(0)=\frac{1}{b}$$ Find an example of a function $f(x,y)$ such that the solution exists only on the interval $(-b,b)$ for $b>0$. I thought to pick $y(x)=\frac{1}{\sqrt{b^2-x^2}}$ since this is defined on $(-b,b)$. Then...
  10. sinkersub

    A: Reciprocal series, B: Laurent Series and Cauchy's Formula

    Problem A now solved! Problem B: I am working with two equations: The first gives me the coefficients for the Laurent Series expansion of a complex function, which is: f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n This first equation for the coefficients is: a_n = \frac{1}{2πi} \oint...
  11. C

    Do Cauchy sequences always converge?

    Hello evry body let be $(u_{n}) \in \mathbb{C}^{\matbb{N}}$ with $u_{n}^{2} \rightarrow 1$ and $\forall n \in \mathbb{N} (u_{n+1) - u_{n}) < 1$. Why does this sequences converge please? Thank you in advance and have a nice afternoon:oldbiggrin:.
  12. Z

    Proof: Every convergent sequence is Cauchy

    Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...
  13. O

    MHB Is Every Sequence with a Cauchy Subsequence Also Cauchy?

    Let $\left\{{x}_{n}\right\}$ be a sequence...İf $\left\{{x}_{2n}\right\}$ is caucy sequence, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence ?
  14. O

    MHB Meir keeler contractive mappings and cauchy sequence

    Denote with ${\varPsi}_{st}$ the family of strictly nondecreasing functions ${\Psi}_{st}:[0,\infty)\to [0,\infty)$ continuous in $t=0$ such that ${\Psi}_{st}=0$ if and only if $t=0$ ${\Psi}_{st}(t+s)\le {\Psi}_{st}(t)+{\Psi}_{st}(s)$. Definition: Let $\left(X,d\right)$ be a metric space and...
  15. B

    Cauchy Schwarz equality implies parallel

    I'm learning about Support Vector Machines and would like to recap on some basic linear algebra. More specifically, I'm trying to prove the following, which I'm pretty sure is true: Let ##v1## and ##v2## be two vectors in an inner product space over ##\mathbb{C}##. Suppose that ## \langle v1 ...
  16. naima

    Things that do not depend on the choice of Cauchy surface

    A Cauchy surface is a 3d spacelike slice of spacetime. Could you read the definition (17) p18 the author says that "one can proove that ##\sigma## does not depend on the choice of the Cauchy surface because the functions obey the law of motion." Could you elaborate? Thanks Could anyone add "h"...
  17. Z

    Calculating integrals using residue & cauchy & changing plan

    Homework Statement \int_{0}^{2\pi} \dfrac{d\theta}{3+tan^2\theta} Homework Equations \oint_C f(z) = 2\pi i \cdot R R(z_{0}) = \lim_{z\to z_{0}}(z-z_{0})f(z) The Attempt at a Solution I did a similar example that had the form \int_{0}^{2\pi} \dfrac{d\theta}{5+4cos\theta} where I would change...
  18. evinda

    MHB Cauchy problem - unique solution

    Hello! (Wave) Theorem: Suppose that the function $f(x,y)$ is continuous as for $x$ in a space $\Omega \subset \mathbb{R}^2$ (not necessarily bounded) and $f$ satisfies the Lipschitz condition as for $y$ for any closed and bounded space $\Omega^1 \subset \Omega$. Then $\forall (x_0, y_0) \in...
  19. O

    MHB To prove that Cauchy sequence

    My Questions: 1) İn both sides of inequality of (*) why we use "n", that is, why we do multiplication with "n" ? 2) in (**) by Letting $n\to\infty$ we obtain $\lim_{{n}\to{\infty}} n\left[d\left({T}^{n}x,{T}^{n+1}x\right)\right]{}^{r}=0$ How this...
  20. A

    MHB Partial Differntial problem Cauchy

    Find surface of $\begin{array}{l} \text{Problem Cauchy} \\ {a^2} \cdot {x_2} \cdot u \cdot {u_{{x_1}}} + {b^2} \cdot {x_1} \cdot u \cdot {u_{{x_2}}} = 2{c^2}{x_1}{x_2}{\rm{ }} \\ \end{array}$ The partial differntial equation passes through ${\rm{ C: = \{ }}\frac{{{x^2}}}{{{a^2}}} +...
  21. Julio1

    MHB Solve Cauchy Problem: Help Needed with Separate Variable Method

    Solve the Cauchy problem: \begin{align} \dfrac{dx}{dr} &= y\\ x(0,s) &= s \end{align} Help please, I don't remember how solve this :(. Separate variable isn't the method?
  22. A

    Convergence and Cauchy Criterion

    Homework Statement Suppose the sequence (xn) satisfies |xn + 1 - xn| < 1/n2, prove that (xn) is convergent. Homework Equations |xn - xm| < ɛ The Attempt at a Solution If m > n, then |xn - xm| < |xn - xn + 1| + |xn + 1 - xn + 2| + ... + |xm - 1 - xm| < 1/n2 + 1/(n+1)2 + ... + 1/(m - 1)2 <...
  23. nuuskur

    Bernoulli differential equation, Cauchy problem

    Homework Statement Observe a Cauchy problem \begin{cases}y' + p(x)y =q(x)y^n\\ y(x_0) = y_0\end{cases} Assume ##p(x), q(x)## are continuous for some ##(a,b)\subseteq\mathbb{R}## Verify the equation has a solution and determine the condition for there to be exactly one solution. Homework...
  24. J

    Derive Euler–Bernoulli equation from Navier-Cauchy equations

    Hello Is it possible to derive the Euler–Bernoulli equation: \frac{d^2}{dx^2} \left(EI \frac{d^2w}{dx^2} \right) = q from Navier-Cauchy equations: \left( \lambda + \mu \right)\nabla\left(\nabla \cdot \textbf{u} \right) + \mu \nabla^2\textbf{u} + \textbf{F} = 0 I don't really know where to...
  25. O

    MHB Cauchy sequence in fixed point theory

    İn some articles, I see something... For example, Let we define a sequence by ${x}_{n}=f{x}_{n}={f}^{n}{x}_{0}$$\left\{{x}_{n}\right\}$. To show that $\left\{{x}_{n}\right\}$ is Cauchy sequence, we suppose that $\left\{{x}_{n}\right\}$ is not a Cauchy sequence...For this reason, there exists a...
  26. O

    MHB Best wishes :)Question: How do we use supremum in the proof of Cauchy sequence?

    Let $\left(E,d\right)$ be a complete metric space, and $T,S:E\to E$ two mappings such that for all $x,y\in E$, $d\left(Tx,Sy\right)\le M\left(x,y\right)-\varphi\left(M\left(x,y\right)\right)$, where $\varphi:[0,\infty)\to [0,\infty)$ is a lower semicontinuous function with...
  27. A

    Why isnt Cauchy's formula used for the perimeter of ellipse?

    So the formula for an ellipse in polar coordinates is r(θ) = p/(1+εcos(θ)). By evaluating L = ∫r(θ) dθ on the complex plane on a circle of circumference ε on the centered at the origin I obtained the equation L = (2π)/√(1-ε^2). Why then does Wikipedia say that the formula for the perimeter is...
  28. ognik

    MHB Proof of Cauchy Integral formula

    Hi, looking at a proof of Cauchy Integral formula, I have (at least) one question, starting from the step below $ \int_{{C}}^{}\frac{f(z)}{z-{z}_{0}} \,dz - \int_{{C}_{2}}\frac{f(z)}{z-{z}_{0}} \,dz = 0 $ , where $ {C}_{2}$ is the smaller path around the singularity at $ {z}_{0} $ Let...
  29. SSGD

    What is this differential equation? I'm going crazy

    I have been working on a math problem and I keep getting the some type of PDEs. x*dU/dx+y*dU/dy = 0 x*dU/dx+y*dU/dy+z*dU/dz = 0 ... x1*dU/dx1+x2*dU/dx2+x3*dU/dx3 + ... + xn*dU/dxn= 0 dU/dxi is the partial derivative with respect to the ith variable. Does anyone know about this type of PDE...
  30. O

    MHB Why Is the Existence of m and n Guaranteed in the Cauchy Sequence Proof?

    to prove that $\left\{{x}_{n}\right\}$ is Cauchy seqeunce we use a method. I have some troubles related to this method. Please help me... $\left\{{c}_{n}\right\}$=sup$\left\{d\left({x}_{j},{x}_{k}\right):j,k>n\right\}$.Then $\left\{{c}_{n}\right\}$ is decreasing. If ${c}_{n}$ goes to 0 as n...
  31. ognik

    MHB Cauchy Integral Theorem with partial fractions

    (Wish there was a solutions manual...). Please check my workings below Show $ \int \frac{dz}{{z}^{2} + z} = 0 $ by separating integrand into partial fractions and applying Cauchy's Integral theorem for multiply connected regions. For 2 paths (i) |z| = R > 1 (ii) A square with corners $ \pm 2...
  32. ognik

    MHB 1st Derivative of Cauchy Integral formula

    Hi - I know the final result for the n'th derivative, I am looking though at getting an expression for the 1st derivative of f(z). From $ f({z}_{0}) = \frac{1}{2\pi i} \oint_{c} \frac{f(z)}{z - {z}_{0}}dz $ we get $ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =...
  33. RJLiberator

    The Cauchy Riemann Equations for this Function

    Homework Statement Verify that each of the following functions is entire: f(z)=(z^2-2)e^(-x)e^(-iy) Homework Equations The Cauchy Riemann equations u(x,y) = ______ and v(x,y) = ______ u_y=-v_x u_x=v_y The Attempt at a Solution So, I've done a few of these problems and understand that to...
  34. N

    Complex Analysis: Use of Cauchy

    http://www.math.hawaii.edu/~williamdemeo/Analysis-href.pdf Please look at problem 2 on page 39 of the problems/solutions linked above. I know I'm going to kick myself when someone explains this to me but how was equation "(31)" of the solution obtained? The first term of the RHS of (31) is...
  35. Z

    Variance and Cauchy Distribution

    Dear all, I'm not a mathematician so please excuse me for a certain lack of strictness ... I work on random signals in physics, these signals are most of the time called "noise" for us. For example, we can speak about x(t), a time domain random signal. Very usually, the statistics...
  36. V

    Complex Gaussian Integral - Cauchy Integral Theorem

    Homework Statement I have to prove that I(a,b)=\int_{-\infty}^{+\infty} exp(-ax^2+bx)dx=\sqrt{\frac{\pi}{a}}exp(b^2/4a) where a,b\in\mathbb{C}. I have already shown that I(a,0)=\sqrt{\frac{\pi}{a}}. Now I am supposed to find a relation between I(a,0) and \int_{-\infty}^{+\infty}...
  37. P

    MHB Construction of a Cauchy sequence

    I need to construct a cauchy sequence ${r}_{n}$ such that its rational for all n belonging to set of natural numbers, but the limit of the sequence is not rational when n tends to infinity. I know that all convergent sequences are cauchy but I can't randomly conjure up a sequence that satisfies...
  38. PsychonautQQ

    Problems understanding proof of Cauchy theorem

    Here is part of the proof via wikipedia: "We first prove the special case that where G is abelian, and then the general case; both proofs are by induction on n = |G|, and have as starting case n = p which is trivial because any non-identity element now has order p. Suppose first that G is...
  39. B

    MHB Complex function that satisfies Cauchy Riemann equations

    Hi, I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this: Find all the complex numbers z∈ C such that f(z)=z cos (z ̅). [z ̅ is z-bar, the complex conjugate). Thanks!
  40. Math Amateur

    MHB Apostal Chapter 4 - Cauchy Sequences - Example 1, Section 4.3, page 73

    I need some help in fully understanding Example 1, section 4.3 Cauchy Sequences, page 73 of Apostol, Mathematical Analysis. Example 1, page 73 reads as follows: https://www.physicsforums.com/attachments/3844 https://www.physicsforums.com/attachments/3845 In the above text, Apostol writes: "...
  41. evinda

    MHB Every Cauchy sequence converges

    Hey! (Wave) Sentence: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. Proof: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show that, without loss of generality, we can suppose that $x_i \in \mathbb{Z}_p$.Let...
  42. X

    What is the proof that the limit of a Cauchy sequence of integers is an integer?

    prove that limit of a cuachy sequence of integers is an integer
  43. C

    Every bounded sequence is Cauchy?

    I've been very confused with this proof, because if a sequence { 1, 1, 1, 1, ...} is convergent and bounded by 1, would this be considered to be a Cauchy sequence? I'm wondering if this has an accumulation point as well, by using the Bolzanno-Weirstrauss theorem. I really appreciate the help...
  44. evinda

    MHB Cauchy Sequences: What it Means to be $|x_{n+1}-x_n|_p< \epsilon$

    Hi! (Wave) I am looking at the following exercise: If $\{ x_n \}$ is a sequence of rationals, then this is a Cauchy sequence as for the p-norm, $| \cdot |_p$, if and only if : $$\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$$ That's what I have tried: $\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$ means...
  45. Fredrik

    Cauchy sequences and absolutely convergent series

    Homework Statement I want to prove that if X is a normed space, the following statements are equivalent. (a) Every Cauchy sequence in X is convergent. (b) Every absolutely convergent series in X is convergent. I'm having difficulties with the implication (b) ⇒ (a). Homework Equations Only...
  46. D

    Are there any metric spaces with no Cauchy sequences?

    A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default? When trying to think of a space with no cauchy...
  47. Darth Frodo

    Complex Analysis: Cauchy Riemann Equations 2

    Hi All, I was reading through Kreyzeig's Advanced Engineering Mathematics and came across two theorems in Complex Analysis. Theorem 1: Let f(z) = u(x,y) + iv(x,y) be defined and continuous in some neighborhood of a point z = x+iy and differentiable at z itself. Then, at that point, the...
  48. DavideGenoa

    Cauchy product with both extremes infinites

    Dear friends, I have been told that if ##\{a_n\}_{n\in\mathbb{N}}##, ##\{a_{-n}\}_{n\in\mathbb{N}^+}##, ##\{b_n\}_{n\in\mathbb{N}}## and ##\{b_{-n}\}_{n\in\mathbb{N}^+}## are absolutely summable complex sequences -maybe even if only one i between ##\{a_n\}_{n\in\mathbb{Z}}## and...
  49. KleZMeR

    Cauchy - Riemann Function in terms of Z

    Homework Statement I found the function V, which is the conjugate harmonic function for U(x,y)=sin(x)cosh(y). I am attaching my work. It turns out to be a two-term function with trig factors. I am then to write F(Z) in terms of Z, but is plugging in x, and y, in terms of Z into my trig...
  50. K

    Gradient Descent and Cauchy Method in Differential Equations

    http://www.math.uiuc.edu/documenta/vol-ismp/40_lemarechal-claude.pdf I don't understand why we use theta for equation (1) Θ>0 but why α=-θX? Thanks.
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