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. K

    Proof of convergence by proving a sequence is Cauchy

    Homework Statement Let 0 < r < 1. Let {A_n} be a sequence of real numbers such that |A_n+1 - A_n| < r^n for all naturals n. Prove {A_n} converges. Homework Equations A sequence of real numbers is called Cauchy, if for every positive real number epsilon, there is a positive integer N...
  2. K

    Is xn+yn a Cauchy sequence if xn and yn are Cauchy sequences?

    Homework Statement Let xn and yn be Cauchy sequences. Give a direct argument that xn+yn is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem. Homework Equations The Attempt at a Solution given epsilon>0 there exists an N in the natural...
  3. M

    Prove Cauchy Sequence Convergence: {p_n} \rightarrow p

    Homework Statement Suppose that {p_n} is a Cauchy sequence and that there is a subsquence {p_{n_i}} and a number p such that p_{n_i} \rightarrow p. Show that the full sequence converges, too; that is p_n \rightarrow p. Homework Equations The Attempt at a Solution Take \varepsilon...
  4. P

    Cauchy theorem and Wick rotation

    Homework Statement Hi! I have a little problem understanding a proof ('Wick rotation') which I found in a textbook on QFT. Assume: f and g are polynomials with degree(g)-degree(f) >= 2 and f/g has no poles on the closed 1. and 3. quadrant. proposition...
  5. J

    Triangle Inequality and Cauchy Inequality Proofs

    Homework Statement The question says to find a proof for Cauchy's Inequality and then the Triangle Inequality. This is an elementary linear algebra class I'm doing, so I can't use inner products or anything. Homework Equations The Attempt at a Solution I got the proofs using algebra, but I'm...
  6. M

    Cauchy Intergral Formula sin(i)?

    Cauchy Intergral Formula sin(i)?? Homework Statement Circle of radius 2 centered at the origin oriented anticlockwise. Evaluate: \int\frac{sin(z)}{z^{2} +1}Homework Equations I think I'm supposed to be using the Cauchy Integral Formula, so \int\frac{f(z) dz}{z - z_{0}} = 2\piif(z_{0})The...
  7. J

    Extended Real definition of Cauchy sequence?

    Is there an extended definition of a Cauchy sequence? My prof wants one with a proof that a sequence divergent to infinity is Cauchy and vice versa. My first thought was that a sequence should be Cauchy if it is Cauchy in the real sense or else that for any M, there are nth and mth terms of...
  8. Fredrik

    Equivalence classes of Cauchy sequences

    \mathbb R can be defined as "any (Dedekind-)complete ordered field". This type of abstract definition is a different kind than e.g. the "equivalence classes of Cauchy sequences" construction. I prefer abstract definitions over explicit constructions, so I would be interested in seeing similar...
  9. ╔(σ_σ)╝

    Cauchy sequence; I need some help

    Homework Statement Let x_{n} be a Cauchy sequence. Suppose that for every \epsilon>0 there is n > \frac{1}{\epsilon} such that |x_{n}| < \epsilon. Prove that x_{n} \rightarrow 0.Homework Equations The Attempt at a Solution My problem with the question is I do not understand it. if, |x_{n}|...
  10. S

    Solving Cauchy Problem: General Solution of xy3zx+x2z2zy=y3z

    Homework Statement getting gen sol of xy3zx+x2z2zy=y3z solve cauchy problem x=y=t, z=1/t The Attempt at a Solution i got gen sol F(C1,C2)=0 as C1=x/z, C2=y4-x2z2 i inserted t for x and y and 1/t for z and ended up with C1-2=1/(C22) I'm unsure what to do from...
  11. Z

    Is Using Distribution Theory Overkill for Differentiating Under the Integral?

    would it be valid (in the sense of residue theorem ) the following evaluation of the divergent integral ? \int_{-\infty}^{\infty} \frac{dx}{x^{2}-a^{2}}= \frac{ \pi i}{a} also could we differentiate with respect to a^{2} inside the integral above to calculate...
  12. M

    Cauchy sequences, induction, telescoping property

    Homework Statement Scanned and attached Homework Equations I am guessing a combination of induction and the telescoping property. The Attempt at a Solution I'm studying this extramurally, and I've just hit a wall with this last chunk of the sequences section, so if someone can...
  13. P

    Null Hyperplanes & Cauchy Surfaces in Spacetimes

    Is a null hyperplane a Cauchy surface in Minkowski spacetime? What in case of other spacetimes?
  14. D

    Cauchy -schwarz inequality help

    Need help proving Cauchy Schwarz inequality ... the first method I know is pretty easy \displaystyle\sum_{i=1}^n (a_ix-b_i)^2 \geq 0 expanding this and using the discriminatant quickly establishes the inequality..The 2nd method I know is I think a easier one , but I don't have a clue about...
  15. R

    Derivatives of Cauchy Distribution

    Hi guys, I would like to ask you where you spot the mistake in the derivatives of the loglikelihood function of the cauchy distribution, as I am breaking my head :( I apply this to a Newton optimization procedure and got correct m, but wrong scale parameter s. Thanks! LLF =...
  16. T

    Does the Series 1/(n * Log(n)) Converge Using the Cauchy Condensation Test?

    Homework Statement I need to determine, using the Cauchy Condensation Test, whether or not the series 1/(n * Log(n)) converges. Homework Equations The Attempt at a Solution I believe that this series converges iff 2^n(1/(2^n*Log(2^n)) converges (Cauchy Condensation Test). I...
  17. T

    Proof of Non-Cauchy Sequence: s_n = 1 + 1/2 + ... + 1/n

    Homework Statement For each n \in N, let s_n = 1 + 1/2 + ... + 1/n. By considering s_2n - s_n, prove that {s_n} is not Cauchy. Homework Equations The Attempt at a Solution I know that s_2n - s_n = (1 + 1/2 + ... + 1/n + 1/(n+1) + ... + 1/2n) - (1 + 1/2 + ... + 1/n)...
  18. T

    Proof of Cauchy Sequence Convergence with Subsequence

    Homework Statement If {s_n} is a Cauchy sequence of real numbers which has a subsequence converging to L, prove that {s_n} itself converges to L. Homework Equations The Attempt at a Solution I know that all Cauchy sequences are convergent, and I know that any subsequences of a...
  19. estro

    Series convergence and Cauchy criterion

    The Attempt at a Solution * forgot to state that I choose m > n > max { N_1, N_2 }. I'm not sure if i did it right, but seems ok to me =) Will appreciate your opinion...
  20. S

    Cauchy real and dedekind real are equivalent or isomorphic

    Hiya, I am looking for the proof for cauchy real and dedekind real are equal (isomorphic). I know they are not equal (CR \= DR) but I need to prove them point to the same real number or mapping from CR -> DR, DR -> CR. I have looked at the textbooks on number system, real analysis and calculus...
  21. D

    Proving Cauchy Sequences with Cosine Function

    Homework Statement Well, my problem is proving that sequences are in fact Cauchy sequences. I know all the conditions that need to be satisfied yet I cannot seem to apply it to questions. (Well, only the easy ones!) My question is, prove that X_{n} is a Cauchy sequence, given that...
  22. K

    Domain of solution to Cauchy prob.

    Prove that the solution of the CP y'=-(x+1)y^2+x y(-1)=1 is globally defined on all of \mathbb{R} How would you go about this? I thought about studying the sign of the right member if the equation. But what would I do next?
  23. K

    Solving Cauchy Prob: y'=sin(x+y+3) y(0)=-3

    y'=\sin (x+y+3) y(0)=-3 I tried substituting x+y+3=u and solving I get \tan (u(x)) - \sec (u(x)) = x but what the heck can I do now?
  24. B

    Cauchy Boundary Conditions on a Wave

    Homework Statement So using the D'Alembert solution, I know the solution of the wave equation is of the form: y(x,t) = f(x-ct) + g(x+ct) I'm told that at t=0 the displacement of an infinitely long string is defined as y(x,t) = sin (pi x/a) in the range -a<= x <= a and y =0...
  25. M

    Complex Analysis: Cauchy Integral Formula

    Homework Statement The problem, for reference, is from Sarason's book "Complex Function Theory, 2nd edition" and is on page 81, Exercise VII.5.1. Let C be a counterclockwise oriented circle, and let f be a holomorphic function defined in an open set containing C and its interior. What is...
  26. J

    Cauchy Integral Formula Problem

    Homework Statement \oint \frac{dz}{z^2 + z} = 0, C: abs(z) > 1 Homework Equations \oint \frac{f(z)}{z-z_0} dz = 2i\pi * f(z_0) The Attempt at a Solution Under normal circumstances, I usually deal with these in the following way. I say that F(z) = 1 (the value in the...
  27. B

    Integrating around contour (Cauchy)

    Homework Statement The question asks that you prove that \int\frac{sin^{2}x}{x^2}dx = \pi / 2 The integral is from zero to infinity, but I don't know how to add those in latex. Homework Equations Use a contour integral to get around the pole at z = 0. The problem is, I'm really really foggy...
  28. S

    Proof of Cauchy Criterion for Riemann Integrals

    Homework Statement Some proofs I've looked at vary, but they generally follow the format show here: http://en.wikibooks.org/wiki/Real_Analysis/Riemann_integration#Theorem_.28Cauchy_Criterion.29 This isn't a question about an exercise, but rather a request for a clarification or a way of...
  29. M

    Worked out examples using Cauchy criterion for series

    Hello everyone, Can anybody suggest a website that has worked out examples using the Cauchy Criterion for Series? or, if your feeling ambitious, work out the following problems below: 1. \sum^{\infty}_{n=1}1/n 2. \sum^{\infty}_{n=1}1/(n(n+1))The reason why I'm asking for this is because our...
  30. D

    Cauchy Schwarz Proof Question

    Homework Statement Prove that |x'y| <= ||x|| ||y|| for vectors x, y Homework Equations ||x|| is the norm of x x' is the transpose of x The Attempt at a Solution ||(x/||x||)-(y/||y||)|| = [ (x'/||x|| - y'/||y||)(x/||x|| - y/||y||) ]^1/2 = [-1/(||x||*||y||) (x'y + y'x) +2]^1/2 we...
  31. K

    Cauchy sequence & Fixed point

    Cauchy sequence & "Fixed" point Homework Statement Suppose that f: Rd->Rd and there is a constant c E (0,1) such that ||f(x)-f(y)|| ≤ c||x-y|| for all x, y E Rd. Let xo E Rd be an arbitrary point in Rd, let xn+1=f(xn). Prove that a) f is continuous everywhere. b) (xn) is Cauchy. c) (xn)...
  32. G

    Complete by taking an arbitrary cauchy sequence

    Homework Statement (1) Prove the space \ell_\infty is complete (2)In \ell_\infty(R) , let Y be the subspace of all sequences with only finitely many non-0 terms. Prove that Y is not complete. The Attempt at a Solution (1)I can show that \ell\infty is complete by taking an arbitrary...
  33. G

    Completeness of R2 with Taxicab Norm

    Homework Statement Given R is complete, prove that R2 is complete with the taxicab norm The Attempt at a Solution you know that ,xk \rightarrow x , yk \rightarrow y Then, given \epsilon, choose Nx and Ny so that \left|x_n - x_m\left| and \left|y_n - y_m\left| are less than...
  34. B

    Cauchy Sequence in Metric Space

    For a metric space (X,d), prove that a Cauchy sequence {xn} has the property d(xn-xn+1)--->0 as n--->\infty In working this proof, is it really as simple as letting m=n+1?
  35. K

    What is the Significance of the Cauchy Integral Theorem in Complex Analysis?

    Could someone tell me what there is so astonishing about the Cauchy integral theorem? No that I doubt that it is, I simply and obviously do not understand it fully. My main issue is that a closed real line integral naturally gives zero and so no big deal that what happens in the complex case. So...
  36. H

    How to Use Cauchy Integral Formula for Circle Contour Integrals?

    Homework Statement Using the Cauchy Integral Formula compute the following integrals,where C is a circle of radius 2a centered at z=o, where 2a<pi Homework Equations \oint\frac{(z-a)e^{z}}{(z+a)sinz} The Attempt at a Solution
  37. K

    Cauchy sequence with a convergent subsequence

    Homework Statement Theorem: In a metric space X, if (xn) is a Cauchy sequence with a subsequence (xn_k) such that xn_k -> a, then xn->a. Homework Equations N/A The Attempt at a Solution 1) According to this theorem, if we can show that ONE subsequence of xn converges to a, is that...
  38. K

    Proving (i) for Cauchy Sequences in Completeness Theorem

    Homework Statement Least Upper Bound (LUB) Principle: every nonempty subset S of R that is bounded above has a least upper bound. Completeness Theorem: every Cauchy sequence of real numbers converges. So R is complete. To prove that Completness Theorem implies the least upper bound...
  39. F

    Solving Cauchy Residue Theorem for p(t) in Complex Analysis Homework

    Homework Statement p(t) = integral[-inf,+inf] ( x/sinh(x) exp (i t x) dx) Homework Equations singularity @ x = n*pi*i where n = +-1, +-2, +-3,... Near n*pi*i one can write sinh(x) ~ (x - n*pi*i) The Attempt at a Solution I apply the cauchy residue theorem. For a positive...
  40. S

    The cauchy problem and the equations

    Hi everyone! again d'inverno! to tell the truth I don't really understand what is going on in the cauchy problem! 1) in section 13.5 "the cauchy problem", it is said that the field equations can be written as the forms in 13.12 to 13.14 can anyone tell me how? actually I tried to use...
  41. K

    Mean value theorem & Cauchy sequence

    Homework Statement Let a0=0 and an+1=cos(an) for n≥0. a) prove that a2n≤a2n+2≤a2n+3≤a2n+1 for all n≥0. b) use mean value theorem to find a number r<1 such that |an+2-an+1| ≤ r|an-an+1| for all n≥0. Using this, prove that the sequence {an} is Cauchy. Homework Equations N/A The Attempt...
  42. K

    Every Cauchy sequence of real numbers converges

    Homework Statement I understand everything except the last two lines. I am really confused about the last two lines of the proof. (actually I was never able to fully understand it since my first year calculus) I agree that if ALL three of the conditions n≥N, k≥K, and nk≥N are satisfied...
  43. K

    Definition of Cauchy Sequence

    "Definition: A sequence of real numbers (an) is Cauchy iff for all ε>0, there exists N s.t. n≥N and m≥N => |an-am|<ε. An equivalent definition is: for all ε>0, there exists N s.t. n≥N => |an-aN|<ε. " ============================================= I don't exactly see why these definitions...
  44. N

    Does Cauchy Test Fail If $\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0$?

    Homework Statement True Or False if f(x) continuous in [a,\infty] and \lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 Then \int_a^\infty f(x)dx converge Homework Equations Anything from calc 1 and 2 The Attempt at a Solution Actually I'm really stuck.. My main motive is to try and...
  45. L

    Cauchy sequence without limit in a complete space?

    I know I'm doing something wrong here, but I can't find my mistake. 1) R^2 is a complete metric space under the ordinary Euclidean metric. 2) Consider the circle of radius 2, centered at the origin in R^2. 3) Construct a sequence {x_n} as follows: x_1 is at the apex of the circle (0,2)...
  46. R

    Cauchy Integral Extension Complex Integrals

    Homework Statement Allow D to be the circle lz+1l=1, counterclockwise. For all positive n, compute the contour integral. Homework Equations int (z-1/z+1)^n dz The Attempt at a Solution I know to use the extension of the CIF. Where int f(z)/(z-zo)^n+1 dz = 2(pi)i*...
  47. Z

    Cauchy trick for divergent integrals.

    is this trick valid at least in the 'regularization' sense ?? for example \int_{-\infty}^{\infty} \frac{dx}{x^{2}-a^{2}} then we replace thi integral above by \int_{-\infty}^{\infty} \frac{dx}{x^{2}+ie-a^{2}} for 'e' tending to 0 using Cauchy residue theorem i get...
  48. D

    Procedure to find Cauchy Integral

    Homework Statement I have a question - just to check when we know the whole function is not analytic at some point of z. We can use cauchy integral formula of 2*pi*j*f(a) to find the answer. In between; one of such method is to use Partial Fraction to break up the rational functions. So...
  49. N

    Query on Cauchy Riemann Condition question

    Dear Friends and Colleagues! I have this practise question:- Show that z(sin(z))(cos(z)) statisfies the Cauchy-Riemann Conditions for analyticity for all values of z. Does 1/[z(sin(z))(cos(z))] statisify simiar conditions? Calculate the derivative of 1/[z(sin(z))(cos(z))] at z=0, +...
  50. J

    Cauchy sequences and uniform convergence

    Homework Statement Suppose the infinite series \sum a_v is NOT absolutely convergent. Suppose it also has an infinite amount of positive and an infinite amount of negative terms. Homework Equations The Attempt at a Solution Say we want to prove it converges by proving...
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