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. L

    A Going from Cauchy Stress Tensor to GR's Energy Momentum Tensor

    Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units? Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR? If so, What...
  2. U

    Complex Integration Along Given Path

    From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...
  3. H

    Learning to use the Cauchy criterion for infinite series

    ##s_1=2## ##s_2=4## ##s_3=5.333## ##s_4=5.9999## ##(s_n)## is increasing, but unable to guess a bound. Let's see if Cauchy criterion can do something. For n>2, $$ s_{n+k} - s_n = \frac{2^{n+1} }{(n+1)!} + \frac{ 2^{n+2} }{(n+2)!} + \cdots \frac{2^{n+k} }{(n+k)!} $$ $$ s_{n+k} - s_n <...
  4. PeterDonis

    A Can a Null Surface Satisfy the Requirements for a Cauchy Surface?

    The definition of a Cauchy surface, as given in, for example, Wald Section 8.3, is "a closed achronal set ##\Sigma## for which ##D(\Sigma) = M##", i.e., every past and future causal curve (timelike or null) through any point in the entire spacetime intersects ##\Sigma##. The definition of...
  5. H

    If ##|s_{n+1} - s_n| \lt 1/2^n##, then ##(s_n)## is a Cauchy sequence

    My attempt: It can be proved that ##\lim \frac{1}{2^n} = 0##. Consider, ##\frac{\varepsilon}{k} \gt 0##, there exists ##N##, such that $$ n \gt N \implies \frac{1}{2^n} \lt \varepsilon $$ Take any ##m,n \gt N##, and such that ##m - k = n##. ##|s_m - s_{m-1} | \lt \frac{1}{2^{m-1}} \lt...
  6. D

    I Cauchy Stress Tensor in Applied Strength of Materials

    I am in a course in applied strength of materials and we often use the 3D stress tensor for stress analysis of materials i.e. Mohr's circles, bending, torsion, etc. Is the stress-energy tensor in relativity basically a 4-d extension to the Cauchy stress tensor commonly used in mechanical...
  7. ergospherical

    I Show Maxwell's Eqns. on a Cauchy Surface (Wald Ch. 10 Pr.2)

    This problem is Wald Ch. 10 Pr. 2.; it asks us to show that ##D_a E^a = 4\pi \rho## and ##D_a B^a = 0## on a spacelike Cauchy surface ##\Sigma## (with normal vector ##n^a##) of a globally hyperbolic spacetime ##(M, g_{ab})##. Using the expression ##E_a = F_{ab} n^b## for the electric field gives...
  8. F

    Derivative of the deformation gradient w.r.t Cauchy green tensor

    What's the derivative of deformation gradient F w.r.t cauchy green tensor C, where C=F'F and ' denotes the transpose?
  9. R

    Cauchy Riemann complex function real and imaginary parts

    Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...
  10. C

    Showing continuous function has min or max using Cauchy limit def.

    Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##. Proof: First we'll regard the case ## l = \infty ## ( the case...
  11. D

    How Does Lorentz's Equation Transform into Cauchy's Law?

    Hello fellow physicists, I need to prove that when ##\omega << \omega_0##, Lorentz equation for refractive indexes: ##n^2(\omega) = 1 + \frac {\omega^2_p} {\omega^2_0 - \omega^2}## turns into Cauchy's empirical law: ##n(\lambda)=A+\frac B {\lambda^2}## I also need to express A and B as a...
  12. B

    Do Cauchy Sequences Imply Convergent Differences?

    I've started by writing down the definitions, so we have $$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$ $$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w...
  13. J

    Confused with this proof for the Cauchy Schwarz inequality

    Im confused as finding the minimum value of lambda is an important part of the proof but it isn't clear to me that the critical point is a minimum
  14. yucheng

    Understanding the Use of Min in Cauchy Sequences

    I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...
  15. yucheng

    Proof that two equivalent sequences are both Cauchy sequences

    Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##. A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##. We proceeded by...
  16. B

    Is the Function Analytic? Testing the Cauchy Riemann Equations

    I tested the first function with the Cauchy Riemann equations and it seemed to fail that test, so I don't believe that function is analytic. However, I'm really not sure how to show that it is or is not analytic using the definition of the complex derivative.
  17. A

    MHB How can I find the Cauchy principal value of this integral?

    How can I find Cauchy principal value. of this integral \[ n(x) = \int_{a}^{b} \frac{d \omega}{\omega ' ^2 - x^2} \] Where $ a<x<b $ I case $a = 0, b = 3, x = 1$ We get \[ n(1) = \int_{0}^{3} \frac{d \omega}{\omega ' ^2 - 1^2} = −0.3465735902799727 \] The result shown is the Cauchy...
  18. C

    I Cauchy Integral Formula with a singularity

    Dear Everyone, I am wondering how to use the integral formula for a holomorphic function at all points except a point that does not exist in function's analyticity. For instance, Let f be defined as $$f(z)=\frac{z}{e^z-i}$$. F is holomorphic everywhere except for $$z_n=i\pi/2+2ni\pi$$ for all...
  19. C

    MHB Using Cauchy Integral Formula for Laurent Series Coefficients

    Dear Everyone, I am wondering how to use the integral formula for a holomorphic function at all points except a point that does not exist in function's analyticity. For instance, Let $f$ be defined as $$f(z)=\frac{z}{e^z-i}$$. $f$ is holomorphic everywhere except for $z_n=i\pi/2+2ni\pi$ for...
  20. Neothilic

    I How to prove the Cauchy distribution has no moments?

    How can I prove the Cauchy distribution has no moments? ##E(X^n)=\int_{-\infty}^\infty\frac{x^n}{\pi(1+x^2)}\ dx.## I can prove myself, letting ##n=1## or ##n=2## that it does not have any moment. However, how would I prove for ALL ##n##, that the Cauchy distribution has no moments?
  21. Euler2718

    I Finding a non-convergent Cauchy sequence

    Define a metric on ##\mathbb{R}[x]## for distinct polynomials ##f(x),g(x)## as ##d(f(x),g(x)) = \frac{1}{2^{n}}##, where ##n## is the largest positive integer such that ##x^{n}## divides ##f(x)-g(x)##. Equivalently, ##n## is the multiplicity of the root ##x=0## of ##f(x)-g(x)##. Set...
  22. Math Amateur

    MHB Operator Norm and Cauchy Sequence .... Browder, Proposition 8.7 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and...
  23. Math Amateur

    MHB Understanding Theorem 3.4: Mean Value Theorem & Cauchy Riemann Equations

    I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 3.2 The Cauchy Riemann Equations ... I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its...
  24. m4r35n357

    I Cauchy product of several series

    I am trying to make sense of the wikipedia article section regarding Cauchy product of several series. but am stuck right at the start because the notation used there is unfamiliar to me and not explained previously in the article. The commas in ##\Sigma a_1, k_1## etc. mean nothing to me. Am I...
  25. V

    Is the Cauchy momentum equation the general form of Newton II?

    Is it appropriate to say that within classical physics the general form of Newton II is the Cauchy momentum equation? This equation applies to an arbitrary continuum body. Therefore it is more general than the common form of Newton II which applies basically to point masses and centers of mass...
  26. SamRoss

    I Necessity of absolute value in Cauchy Schwarz inequality

    Reading The Theoretical Minimum by Susskind and Friedman. They state the following... $$\left|X\right|=\sqrt {\langle X|X \rangle}\\ \left|Y\right|=\sqrt {\langle Y|Y \rangle}\\ \left|X+Y\right|=\sqrt {\left({\left<X\right|+\left<Y\right|}\right)\left({\left|X\right>+\left|Y\right>}\right)}$$...
  27. C

    I When and How Does an Integral Have a Cauchy Principal Value?

    Under what conditions does an integral have a cauchy principal value and how is it related to an integral having an integrable singularity? E.g $$p.v \int_{-\delta}^{\delta} \frac{dz}{z} = 0$$ If I evaluate the integral along a semi circle in the complex plane I'll get ##i \pi##. So the cauchy...
  28. L

    A Same open sets + same bounded sets => same Cauchy sequences?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
  29. Euler2718

    Showing a sequence of functions is Cauchy/not Cauchy in L1

    Homework Statement Determine whether or not the following sequences of real valued functions are Cauchy in L^{1}[0,1]: (a) f_{n}(x) = \begin{cases} \frac{1}{\sqrt{x}} & , \frac{1}{n+1}\leq x \leq 1 \\ 0 & , \text{ otherwise } \end{cases} (b) f_{n}(x) = \begin{cases} \frac{1}{x} & ...
  30. F

    Showing Uniform Convergence of Cauchy Sequence of Functions

    Homework Statement Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##. Homework Equations Uniform convergence: for all ##\varepsilon >...
  31. S

    MHB Problems for Cauchy Integral Formula

    Hello everyone! I am currently stuck at the two type of questions below, because I am not really sure what method should be used to calculate these question... Could you give me a hint how to do these questions? :(
  32. T

    I Intuition - Cauchy integral theorem

    So folks, I'm learning complex analysis right now and I've come across one thing that simply fails to enter my mind: the Cauchy Integral Theorem, or the Cauchy-Goursat Theorem. It says that, if a function is analytic in a certain (simply connected) domain, then the contour integral over a simple...
  33. E

    I Spivak's proof of Cauchy Schwarz

    I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality. Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result. The problem is...
  34. hassouna

    What is the result of multiplying a vector by its complex conjugate?

    I found that the equation is expressed by there is outer product ...what I really don't get it is if j is a vector then the outer product of j and j is is obtained by multiplying each element of j by the complex conjugate of each element of j which is basically a matrix not a vector
  35. T

    Regarding Real numbers as limits of Cauchy sequences

    Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...
  36. J

    I Interpretation of the Fourier Transform of a Cauchy Distribution

    Hi, I'm struggling with a conceptual problem involving the Fourier transform of distributions. This could possibly have gone in Physics but I suspect what I'm not understanding is mathematical. The inverse Fourier transform of a Cauchy distribution, or Lorentian function, is an exponentially...
  37. L

    Inner Product, Triangle and Cauchy Schwarz Inequalities

    Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
  38. Math Amateur

    MHB Cauchy Sequences and Completeness in R^n ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.6.5 (Completeness of \mathbb{R}^n) ... Duistermaat and Kolk"s Theorem 1.6.5 and its...
  39. Peter Alexander

    Understanding the Cauchy Integration Formula for Analytic Functions

    Hello everyone! I'm having a bit of a problem with comprehension of the Cauchy integration formula. I might be missing some key know-how, so I'm asking for any sort of help and/or guideline on how to tackle similar problems. I thank anyone willing to take a look at my post! Homework Statement...
  40. J

    Cauchy Integral of Complex Function

    Homework Statement Homework Equations Using Cauchy Integration Formula If function is analytic throughout the contour, then integraton = 0. If function is not analytic at point 'a' inside contour, then integration is 2*3.14*i* fn(a) divide by n! f(a) is numerator. The Attempt at a Solution...
  41. Kushwoho44

    Understanding the Cauchy Stress Tensor: Clearing Up My Confusion

    I have been trying to fully grasp the concept of the Cauchy stress tensor and so I thought I'd make a post where I clear up my confusion. There may be subsequent replies as I pose more questions. I am specifically confused at how the stress tensor relates to the control volume in the image...
  42. Math Amateur

    MHB How does Conway's Theorem 2.29 Derive the Simplified Expression for f(z)?

    I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter III Elementary Properties and Examples of Analytic Functions ... Section 2: Analytic Functions ... ... I need help in fully understanding aspects of Theorem 2.29 ...
  43. L

    Cauchy Stress Components from Surface Force

    Hi, I've been trying to figure out how to get Cauchy Stress Tensor components (~9) from a surface force for a while now. My background in this subject is not too deep, but I'm trying to build a renderer simulation in my free time. I can get surface traction from a Stress Tensor: t =...
  44. A

    I Event and Cauchy horizons for a charged black hole

    Consider the Reissner-Nordstrom metric for a black hole: $$ds^{2} = - f(r)dt^{2} + \frac{dr^{2}}{f(r)} + r^{2}d\Omega_{2}^{2},$$ where $$f(r) = 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.$$ We can write $$f(r) = \frac{1}{r^{2}}(r-r_{+})(r-r_{-}), \qquad r_{\pm} = M \pm \sqrt{M^{2}-Q^{2}}.$$ Then...
  45. F

    Prerequisites to Cauchy Integral Formula

    Homework Statement Calculate the integrals of the following functions on the given paths. Why does the choice of path change/not change each of the results? (c) f(z) = exp(z) / z(z − 3) https://www.physicsforums.com/file:///page1image10808 i. a circle of radius 4 centred at 0. ii. a circle...
  46. Mr Davis 97

    I Find Cauchy Principal Value of Improper Integral

    Say I am given the integral ##\displaystyle \int_0^{\infty} \frac{x \sin (ax)}{x^2 - b^2} dx##. How can I determine whether this improper integral converges in the normal sense, or whether I should just look for the Cauchy Principal Value?
  47. mercenarycor

    Contour integration with a branch cut

    Homework Statement ∫-11 dx/(√(1-x2)(a+bx)) a>b>0 Homework Equations f(z0)=(1/2πi)∫f(z)dz/(z-z0) The Attempt at a Solution I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed...
  48. binbagsss

    Inequality quick question context cauchy fresnel integral

    Homework Statement please see attached, I am stuck on the second inequality. Homework Equations attached The Attempt at a Solution I have no idea where the ##2/\pi## has come from, I'm guessing it is a bound on ##sin \theta ## for ##\theta## between ##\pi/4## and ##0## ? I know ##sin...
  49. Cocoleia

    Second order non homogeneous ODE, IVP

    Homework Statement I need to solve: x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9 Homework EquationsThe Attempt at a Solution I know that the answer is: y=x^2+2x^3+x^3lnx Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...
  50. K

    MHB Test for Cauchy sequence (with limsup and log)

    If $\{x_n\}_{n \ge 1}$ is real sequence and $\limsup\limits_{n \to \infty} \frac{1}{n} \log |x_{n+1}-x_n|<0$, prove that $\{x_n\}$ is Cauchy sequence. My work: Let $a=\limsup\limits_{n \to \infty} \frac{1}{n} \log |x_{n+1}-x_n| <0$. Then, for every $\varepsilon >0$ there exist $N \in...
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