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. B

    Cauchy product of series

    I don't understand a small part in the proof that two absolutely convergent series have absolutely convergent cauchy product. Instead of writing the whole thing, I'll write the essentials and the step I'm having trouble with. \sum_{r=1}^{\infty}a_{r} and \sum_{r=1}^{\infty}b_{r} are positive...
  2. Z

    Can Cauchy's residue theorem be used for functions with poles at infinity?

    my question is , let us have the following complex integral \oint f(z)dz where f(z) has a simple pole at z=\infty then by Residue theorem \oint f(z)dz =2\pi i Res(z,\infty,f(z) or equal to the limit (z-\infty )f(z) with 'z' tending to infinity
  3. E

    Proving Cauchy Convergence with Inequalities

    Homework Statement Homework Equations See picture in 1. The Attempt at a Solution See picture in 1. I think what's tripping me up is that I'm not sure how to go about picking my N. I want to show the sequence has a limit by proving that it is Cauchy.
  4. Z

    Cauchy theorem for integrals

    i want to perform the following integrals \int_{-\infty}^{\infty}dx \frac{f(x)}{x^{2}-a^{2}} the problem is that the integral has poles at x=a and x=-a , could we apply i think this is the definition of Hadamard finite part integral, performing an integral with singularities by means...
  5. J

    What is the mistake in my reasoning for Cauchy sequences?

    As far as I understand, a sequence converges if and only if it is Cauchy. So say for some sequence a_n and for all epsilon greater than zero we have |a_n - a_{n+1}| < \epsilon for large enough n. We could then say a_n converges if and only if \lim_{n \rightarrow \infty} a_n - a_{n+1} = 0 ...
  6. L

    Solve Cauchy Integral on Unit Circle: f'(z)/(z-zo) = f(z)/(z-zo)^2

    A. Homework Statement f is analytic inside and on a simple closed contour C and z0 isn't on C. Show: \int f'(z)dz/ (z- zo) = \int f(z)dz/ (z- zo)^2 The Attempt at a Solution \int f'(z)dz/ (z- zo) = 2\pii f'(zo)\int f(z)dz/ (z- zo)^2 = \int [f(z)dz/(z- zo)]/(z-zo) = 2\pii [f(zo)/(zo-zo)] (I...
  7. M

    Prove a Cauchy Sequence using Geometric Sums

    Homework Statement Let {x_n} be a sequence. and let r be a real number 0<r<1. Suppose |x_(n+1) - x_n|<=r|x_n -x_(n-1)| for all n>1. Prove that {x_n} is Cauchy and hence convergent. Homework Equations if |r|<1 then the sequence \sum r^k from k=0 to n converges to 1/(1-r) The...
  8. D

    How do I prove the convergence of a Cauchy sequence using a telescoping sum?

    Homework Statement My problem is this, you are given that for a sequence the following is true: \left|x_{n} - x_{n+1}\right| < \frac{1}{4^{n}} So its an obvious case to prove the sequence is Cauchy. Homework Equations So I'll state the stuff just for the sake of it...
  9. S

    Is a Cauchy Sequence Always Convergent?

    Homework Statement What does it mean when a sequence is Cauchy?
  10. B

    Trig Integral using Cauchy Formula

    Homework Statement Evaluate the integral: I=\frac{1}{2\pi} \int^{2\pi}_0 \frac{d\theta}{1-2aCos\theta + a^2}, 0 < a < 1. This integral is worked out in the book as an example, but I don't understand all the steps. My confusion is highlighted in red below. (From Complex Variables, Stephen...
  11. B

    Cauchy Integral with absolute value in simple curve

    Homework Statement Integrate using Cauchy Formula or Cauchy Theorem: I=\oint_{|z+1|=2}\frac{z^2}{4-z^2} dz (From Complex Variables, Stephen Fisher (2nd Edition), Exercise 2.3.3) Homework Equations \frac{1}{2\pi i}\oint_{\gamma} \frac{f(s)}{s-z} ds= \left\{ \begin{array}{lr}f(z) & : z \in...
  12. C

    Proving Cauchy Sequences: Infinite Subsequences

    Hi, I need to prove that any infinite subsequence {xnk}of a Cauchy sequence {xn}is a Cauchy sequence equivalent to {xn}. My problem is that it seemed way too easy, so I'm concerned that I missed something. Please see the attachment for my solution, and let me know what you think. Thanks.
  13. E

    Cauchy Sequences Triangle Inequality.

    Homework Statement assuming an and bn are cauchy, use a triangle inequality argument to show that cn= | an-bn| is cauchy Homework Equations an is cauchy iff for all e>0, there is some natural N, m,n>=N-->|an-am|<e The Attempt at a Solution I am currently trying to work backwards on this one...
  14. A

    Cauchy Sequence That Does Not Converge In A Space

    Let M be a subspace of l^infinity consisting of all sequences x = (x_j) with at most finitely many nonzero terms. Find a Cauchy sequence in M which does not converge in M, so that M is not complete. Does anyone have any ideas what to use for a Cauchy sequence that does not converge in M? I've...
  15. T

    Webpage title: Solving Inequalities Using the Intermediate Value Theorem

    i got this question http://img412.imageshack.us/img412/3713/88436110xw9.gif here is the solution: http://img297.imageshack.us/img297/6717/14191543qm1.th.gif they are taking the minimal value and the maximal value the innequalitty that the write is correct min< <max but why??
  16. D

    Cauchy Distribution Homework: X/Y Derivation

    Homework Statement http://img132.imageshack.us/img132/1/48572399ly5.png The Attempt at a Solution I tried dividing the two pdf's but that isn't right. If you have X normal and Y normal distributed how can you derive the distribution function of X/Y?
  17. S

    Cauchy Criterion for Series limit

    need to prove there's limit by cauchy criterion http://mathworld.wolfram.com/CauchyCriterion.html the sequence: {X_n}=(10/1)(11/3)...(n+9/2n-1) my try : is that right ? http://img155.imageshack.us/my.php?image=22221wl0.jpg
  18. I

    Sequences limits and cauchy sequences

    Homework Statement prove or refute: if lim(a(2n)-a(n)=o , then a(n) is a cauchy sequence Homework Equations The Attempt at a Solution I need to prove that for every m,n big enough a(m)-a(n)<epsilon so I know for all m and n I can say m=l*n, lim(a(m)-a(n))=lim(a(n*l)-a(n*l/2)...
  19. L

    Cauchy Sequence Problem Update

    This is what it is in my book, i don't know how it went from saying that the difference is 2e and then using e/2 . The part that confuses me , is why the book outlines (1) to then use it in the proof, it looks that (1) doesn't even matter because since e > 0 then e/2 will do just fine. (1)...
  20. L

    Explanation of Cauchy Proof for Convergent Sequence

    Suppose { a_n } converges to A. Choose e > 0, THere is a positive integer N such that, if n, m >= N , then A - e < a_n < A + e and A - e < a_m < A + e Thus for all n, m >= N we find a_n ∈ ( A - e , A + e ) and a_m ∈ ( A - e , A +e ) . the set ( A - e, A +e ) is an interval of length...
  21. L

    Proof of Cauchy Sequence: e/2 Appearing in Proof

    Every convergent sequence is a cauchy sequence Proof: Suppose {a_n} converges to A, choose e > 0 , then e/2 > 0 there is a positive integer N such that n>=N implies abs[a_n - A ] < e/2 the difference between a_n , a_m was less than twice the original choice of e Now if m , n...
  22. L

    Cauchy Sequence Proof: How Does e/2 Appear in the Proof?

    Homework Statement Every convergent sequence is a cauchy sequence Homework Equations Proof: Suppose {a_n} converges to A, choose e > 0 , then e/2 > 0 there is a positive integer N such that n>=N implies abs[a_n - A ] < e/2 the difference between a_n , a_m was less than...
  23. J

    Cauchy integral theorem question

    I recently did a problem in which the integral around a contour contained two residues, the sum of which was zero, so the total integral around the entire path was zero? By the CIT, the function should then be analytic (holomorphic, if you like) inside that contour, but it isn't obviously...
  24. J

    Cauchy sequences and sequences in general

    Is every sequence that converges a Cauchy sequence (in that for every e > 0, there is an integer N such that |a_n - a_m| < e whenever n,m > N)? I think it is because if a sequence a_n converges to L, then you can mark off an open interval of any size about L such that this interval contains all...
  25. G

    Cauchy Sequences: Definition & a(m) Clarification

    By definition, a sequence a(n) has the Cauchy sequence if for eery E>0 ,there exist a natural number N such that Abs(a(n) - a(m) ) < E for all n, m > N Could anyone tell me what is a(m) ? is it a subsequence of a(n) , or could it be any other non related sequence ?
  26. A

    Cauchy Random Variable Homework with Equations and Attempt

    Homework Statement I have attached the problem statement Homework Equations Also find attached The Attempt at a Solution My attempt is attached together with the problem statement and the relevant equations.
  27. C

    Need help with Cauchy Theorum/Formula

    Ok so I am asking for some help, please help. I don't know how to apply the formula given for Cauchy's Theorum, on the complex plane if anyone can show me a good website with examples on this topic, please tell me.. I am struggling and this is like a cry for help, please help. thank you
  28. S

    Show Cauchy Sequence Convergence with sin(n)/2^n

    Homework Statement Show that the following sequence converges: xn= (sin(1)/2) + (sin(2)/2^2) + (sin(3)/2^3) +...+ (sin(n)/2^n) Homework Equations The Attempt at a Solution To show that it converges, i want to show that it is a cauchy sequence (since all cauchy sequences...
  29. U

    How to Show Sequence is Cauchy for n in the Naturals

    For n in the naturals, let p_n = 1 + \frac{1}{2!} + ... + \frac{1}{n!} Show it is cauchy. Attempt: I have set up |p_n+k - p_n | < e , and I have solved for this. I got |p_{n+k} - p_n | = \frac{1}{(n+1)!} + ... + \frac{1}{(n+k)!} I am trying to follow an example in the...
  30. U

    Understanding Cauchy Sequences: Solving for a_n in an Inductive Definition

    Let a_1 and a_2 be arbitrary real number that are not equal. For n \geq 3, define a_n inductively by, a_n = \frac{1}{2} (a_{n-1}+a_{n+2} ) I cannot get the result that the book gets. I proceed, a_{n+1} - a_{n} = \frac{1}{2}(a_n + a_{n-1} ) - \frac{1}{2} (a_{n-1} + a_{n-2} ) =...
  31. Kudaros

    Analysis of an= n1/2: Cauchy Criterion Examined

    Homework Statement Define the Sequence an= n1/2 where n is a natural number. Show that |an+1-an| -> 0 but an is not a cauchy sequence Homework Equations The Attempt at a Solution (Ignore this paragraph)Well, unfortunately I am stuck on the very first part. How exactly do I evaluate the...
  32. P

    A small question about Cauchy sequence

    Homework Statement Let a_n = (1/2)[(1/a_n)+1] and a_1=1, does this sequence converge? Homework Equations A sequence in R^n is convergernt if and only if it's cauchy. A sequence in R^n is called a cauchy sequence if x_k - x_j ->0 as k, j-> infinity. The Attempt at a Solution I...
  33. D

    Understanding Cauchy Sequences in Banach Spaces

    Homework Statement http://img394.imageshack.us/img394/5994/67110701dt0.png Homework Equations A banach space is a complete normed space which means that every Cauchy sequence converges. The Attempt at a Solution I'm stuck at exercise (c). Suppose (f_n)_n is a Cauchy sequence in E. Then...
  34. B

    Cauchy-Riemann Equations and Complex Derivatives: A Homework Problem

    Homework Statement Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write: f'(z)=u_x+iv_x=3x^2 only when z=i Homework Equations Cauchy riemann equations: u_x=v_y , u_y=-v_x f'(z)=u_x+i*v_y The Attempt at a Solution u=x^3 v=(1-y)^3 u_x=3*x^2 v_y=-3*(1-y)^2...
  35. P

    First order pde cauchy problem by method of characteristics

    Ok, so I can get through most of this but I can't seem to get the last part... Here is the problem xU_x + (y^2+1)U_y = U-1; U(x,x) = e^x Characteristic equations are: \frac{dx}{x} = \frac{dy}{y^2+1} = \frac{dU}{U-1} Solving the first and third gives: \frac{U-1}{x} = c_1 The...
  36. P

    First Order PDE Cauchy problem Using Method of Characteristics

    Homework Statement Ok, so I can get through most of this but I can't seem to get the last part... Here is the problem xU_x + (y^2+1)U_y = U-1; U(x,x) = e^x Homework Equations The Attempt at a Solution Characteristic equations are: \frac{dx}{x} = \frac{dy}{y^2+1} =...
  37. M

    Cauchy Riemann & Taylor Expansion.

    Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4. Q1...
  38. J

    Proving Convergence with Cauchy Sequence Method

    Hello all. I'm having trouble on the following homework problem. It seems like it should be easy, but I'm just now sure how to approach it Homework Statement Let (s_n) be a sequence st |s_{n+1} - s_n | < 2^{-n}, \forall n \in \mathbb{N} show that (s_n) converges The Attempt at a...
  39. H

    Cauchy-Schwarz Inequality and Its Relation to Trigonometric Identity

    Homework Statement what is the relation between cauchy_schwarz inequality and this ; Sin(a+b)=Sin(a)Cos(b)+Cos(a)Sin(b):biggrin: Homework Equations The Attempt at a Solution
  40. N

    Complex analysis - Cauchy estimates

    Homework Statement Let D\subset\mathbb{C} be the unitdisc and F=\{f:D\rightarrow D\,|\,\forall z\in D\partial_{\bar{z}}f=0\}, calculate L=\sup_{f\in F}|f''(0)|. Show that there is an g\in F with g''(0)=L. I am a bit stuck. But I think that it might be an idea to start with Cauchy estimate. Any...
  41. A

    Why does the Cauchy integral theorem require the first term to vanish?

    I was looking at the proof of the residue theorem on MathWorld: http://mathworld.wolfram.com/ResidueTheorem.html and got stucked on the 3rd relation. I don't understand why the Cauchy integral theorem requires that the first term vanishes. From the Cauchy integral theorem, the contour...
  42. stevebd1

    Cauchy horizon in a Kerr rotating black hole

    I'm currently in the process of understanding the properties of the Cauchy (or inner event) horizon in a rotating Kerr black hole. Initially, I thought the horizon was a result of centrifugal forces from the rotating ring singularity and extreme frame dragging that forced matter outwards...
  43. R

    Proving Cauchy Sequence: a_n = [a_(n-1) + a_(n-2)]/2

    Homework Statement Prove that the following sequence is Cauchy: a_n = [a_(n-1) + a_(n-2)]/2 (i.e. the average of the last two), where a_0 = x a_1 = y Homework Equations None The Attempt at a Solution I was trying to use the definition of Cauchy (i.e. |a_m - a_n| < e) by...
  44. R

    Solving Problems in a Class Missed: Limits, Isolated Points & Cauchy Sequences

    This is review from class the other day that I managed to miss because of illness and I was wondering if someone could explain how to go about solving these problems: #1 Let B = \left\{ \frac{(-1)^nn}{n+1}:n = 1,2,3,...\right\} Find the limit points of B Is B a closed set? Is B an open set...
  45. R

    Proving Cauchy Sequence of |an-bn|

    I'm basically trying to show that if (an) and (bn) are Cauchy sequences, then (cn) = |an - bn| is also a Cauchy sequence. I know that the triangle inequality is going to be used at one point or another, but I suppose I'm a little confused because: (an) is Cauchy implies |an - am| < e (bn) is...
  46. D

    Cauchy integral theorem

    Hi, so suppose f(z) is a complex function analytic on {z|1<|z|} (outside the unit circle). Also, we know that limit as z-->infinity of zf(z) = A. Now I need to show that for any circle C_{R} centered at origin with radius R>1 and counterclockwise orientation, that \oint f(z)dz = 2\pi iA...
  47. Simfish

    Question on the Cauchy Condensation test

    So it's here http://en.wikipedia.org/wiki/Cauchy_condensation_test My question is this: is the value of the base 2 in 2^k an arbitrary value? Or is there something special about 2? Can we just use something like e^k instead?
  48. V

    What are the properties of the given recursive Cauchy sequence?

    Homework Statement Prove that the sequence x_{n+1} = \frac{(x_{n})^{2} + 2}{2x_{n}} where x_{1} = 1 is Cauchy.Homework Equations The sequence comes from using Newton's Method on x^2 + 2, but we aren't supposed to use sqrt(2) anywhere in the solution. In fact, the next part of the problem is...
  49. L

    Cauchy Condensation Proof

    Hello. This is the presented problem: Suppose (b_{n}) is a decreasing satisfying b_{n}\ge\ 0. Show that the series \sum^{\infty}_{n=1}b_{n} diverges if the series \sum^{\infty}_{n=0}{2^{n}b_{2^{n}}} diverges. I've already proved that i can create \sum^{\infty}_{n=0}{2^{n}b_{2^{n}}} from...
  50. E

    Solving the Euler Cauchy Equation: Finding the General Solution

    Homework Statement Find the general solution of x^2y" - 2y = 0 Homework Equations The Attempt at a Solution Can anyone tell me how to find the general solution of the Euler Cauchy equation. How do we make it into one?? Thanks.
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