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 theorem and fourier transform

    Homework Statement Hi, I have this problem and I don't know how to finish it: Using the Cauchy Theorem, prove that the Fourier tranform of \frac{1}{(1+t^2)} is \pi.e^{-2.\pi.|f|} .( you must show the intergration contour) Stetch the power spectrum. I applied the Fourier transform...
  2. H

    Proving the Cauchy Sequence of (An)

    How could i show that the sequence (An)= (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy? Thanks in advance!
  3. M

    Cauchy Integral Number 2 (Proof)

    Homework Statement If f(z) is analytic interior to and on a simple closed contour C, then all the derivatives f^k(z) , k=1,2,3... exist in the domain D interior to C, and f^k(z)=\frac{k!}{2i\pi}\int_{C}\frac{f(\zeta)d\zeta}{(\zeta-z)^{k+1}} Prove for second derivative. The...
  4. B

    Can Cauchy's Integral Formula Be Used for Contours Larger Than the Given Circle?

    Hello: I am wondering about the following: Let C be the circle |z| = 3. the contour integral g(w) = integral on C of (2*z^2-z-2/z-w)dz can be evaluated by cauchy's integral formula. I am wondering what happens if w is greater than 3. Would you get this: f = 2*z^2-z-2. This is...
  5. B

    Proving the Cauchy Integral Theorem for Complex Analysis

    Hi, I don't understand the cauchy theorem on complex analysis. I have this problem and I would like to have some help for it. The question is: Use the Cauchy Integral Theorem to prove that: \int_{-\infty}^{+\infty}\frac{1}{x^2-2x+5}dx=\frac{\pi}{2} It is told to have a closed...
  6. D

    Proving Cauchy Sequences using the Definition

    Hey all, I've lurked on here and have found you all very useful and I have this question that is really bugging me. How would you prove something is a cauchy sequence using tits definition.
  7. H

    Greens theorem and cauchy theorem help

    I'm doing these in order to prepare for my quiz in a week. I have no clue where to get started or the first step in attempting problem 3 and problem 4. Please do not solve it, I just want a guide and a direction... thanks if you guys don't mind, please download and have a look!
  8. V

    Convergence and Cauchy Sequences in Rational Numbers

    Homework Statement Prove that if {a_{n}} is a sequence of rational numbers such that {a_{n+1}} > {a_{n}} for all n \in \textbf{N} and there exists an M\in \textbf{Q} such that {a_{n}} \leq M for all n \in \textbf{N}, then {a_{n}} is a Cauchy sequence of rational numbers.Homework Equations Do...
  9. Simfish

    Maple How Can I Calculate the Cauchy Sum of a Taylor Polynomial in Maple?

    So... I want to find the Cauchy sum of the Taylor polynomial of \exp x \sin x. I know how to do this with maple, which only requires the command taylor(sin(x)*exp(x), x = 0, n). I can also try the good old f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots...
  10. P

    Cauchy Boundedness: Partial Sums Unbounded?

    Homework Statement Thm: If a sequence is Cauchy than that sequence is bounded. However Take the partial sums of the series (sigma,n->infinity)(1/n). The partial sums form a series which is Cauchy. But the series diverges so the sequence of partial sums is unbounded. Sequence of partial sums...
  11. quasar987

    Analysis - Cauchy caracterisation of completeness

    Homework Statement In my book (Classical Analysis by Marsdsen & Hoffman), they use the monotone bounded sequence property as the completeness axiom. That is to say, they call complete an ordered field in which every bounded monotone sequence converges and they argue that there is a unique (up...
  12. R

    What is a Cauchy surface and its significance in spacetime?

    Would someone care explaining to me what a Cauchy surface is? thanks
  13. H

    Probability plot for Cauchy Distribution

    I have generated 2 columns of normal random variables, Z1 and Z2. Theorectically, Z1/Z2 will follow a Cauchy distribution. The question is, how do I construct a probability plot to show that indeed it is a Cauchy distribution? I tried the follow procedure: -Sort the Z1/Z2 -Rank them and...
  14. D

    Proving Cauchy Sequences with Totient Theorem

    Homework Statement If p does not divide a, show that a_n=a^{p^{n}} is Cauchy in \mathbb{Q}_p. The Attempt at a Solution We can factor a^{p^{n+k}}-a^{p^n}=a^{p^n}(a^{p^{n+k}-1}-1). p doesn't divide a^{p^n} so somehow I must show that a^{p^{n+k}-1}-1 is divisible by larger and larger powers of...
  15. M

    Applying Cauchy Integral Theorem to Compute Integrals over Circular Paths

    Homework Statement For r=1,3,5 compute the following integral: Integral over alpha (e^(x^2)/(x^2-6x)dx Alpha(t) = 2+re^(it) from 0 to 2pi Homework Equations Cauchy Integral Formula: f(z) = 1/(2ipi)Integral over Alpha(f(x)/(x-z)dx) The Attempt at a Solution For r = 1...
  16. L

    Is {q(n) * a(n)} = {p(n) * b(n)} (for all integer n's) a Cauchy Sequence?

    Homework Statement q(n) = Sum(from k=1 to n) 1/n! Exercise 3: Prove that {q(n)}n(forall)Ns is a cauchy sequence. Homework Equations none. The Attempt at a Solution So many attempts at a solution. I know that a sequence is a cauchy sequence if for all epsilons greater than...
  17. W

    Why is the Cauchy Riemann relation important for complex differentiability?

    The cauchy Riemann relations can be written: \frac{\partial f}{\partial \bar{z}}=0 Is there an 'easy to see reason' why a function should not depend on the independent variable [itex]\bar{z}[/tex] to be differentiable?
  18. B

    Proof of Minkowski Inequality using Cauchy Shwarz

    I tried to expand the [SUM{[X sub k + Y sub k]^2}]^1/2 term but I am stuck there.
  19. C

    Proving Cauchy-Schwarz Inequality Using Completing the Square

    Lets say we have: (a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n})^{2} \leq (a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2})(b_{1}^{2} + b_{2}^{2} + ... + b_{n}^{2}) . Let A = a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2} , B = a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n}, C = b_{1}^{2} + b_{2}^{2} + ... +...
  20. G

    Cauchy P.V. of an Improper Integral

    I was doing a Fourier Transform Integral, and was wondering if it would be legitimate for me to choose a semicircle CR on the lower half-plane below the real axis rather than choosing a semicircle CR on the upper half-plane above the real axis. I would expect it to be valid because the contour...
  21. E

    Complex analysis - Cauchy Theorem

    Hi again. Can somebody help me out with this question? "\int_{C_1(0)} \frac {e^{z^n + z^{n-1}+...+ z + 1}} {e^{z^2}} \,dz Where C_r(p) is a circle with centre p and radius r, traced anticlockwise." I'd be guessing that you have to compare this integral with the Cauchy integral formula...
  22. happyg1

    Why Doesn't a Proof by Contradiction Work for Cauchy Convergence?

    Hi, Here's the question: Show that if {x_n} is a cauchy sequence of points in the metric space M, and if {x_n} has a subsequence which converges to x \in M, Prove that x_n itself is convergent to x. Now, I have proved this as follows..I didn't put in all of the details... Let {x_n_k} be the...
  23. S

    Using Cauchy Multiplication to Find Coefficients in Laurent Series for 1/f(z)

    I have a function 2-z^2-2\cos z, which has a zero at z=0. I have determined the Maclaurin series for f: \sum_{j=2}^\infty(-1)^{j-1}\frac{2z^{2j}}{(2j)!}, and now I have to determine the coefficients a_{-j},~\forall j>0, in the Laurent series for a function h, which is defined as...
  24. 1

    Proving the Cauchy Criterion for Sum of Sequences

    How would one prove that the sum of 2 cauchy sequences is cauchy? I said let e>0 and take 2 arbitrary cauchy sequences then |Sn - St|<e/2 whenever n,t>N1 and |St - Sm|<e/2 whenever t,m >N2. So |Sn - Sm|=|Sn - St + St - Sm|<= |Sn - St|+|St - Sm|< e/2 + e/2 <= e So n,m>max{N1, N2}...
  25. M

    What are the criteria for proving equivalence of Cauchy sequences?

    Question: Prove that if a Cauchy sequence x_1, x_2,... of rationals is modified by changing a finite number of terms, the result is an equivalent Cauchy sequence. All the math classes I have taken previously were computational, and my textbook contains almost no definitions. So, I...
  26. M

    Prove Cauchy sequence & find bounds on limit

    Here's the problem statement: Prove that x_1,x_2,x_3,... is a Cauchy sequence if it has the property that |x_k-x_{k-1}|<10^{-k} for all k=2,3,4,.... If x_1=2, what are the bounds on the limit of the sequence? Someone suggested that I use the triangle inequality as follows: let n=m+l...
  27. Oxymoron

    Proving Cauchy Sequences in the p-adic Metric

    Question Consider the sequence \{p^n\}_{n\in\mathbb{N}}. Prove that this sequence is Cauchy with respect to the p-adic metric on \mathbb{Q}. What is the limit of the sequence?
  28. S

    Cauchy Sequence: Understanding the Boundary Condition

    hello all I found this rather interesting suppose that a sequence {x_{n}} satisfies |x_{n+1}-x_{n}|<\frac{1}{n+1} \forall n\epsilon N how couldn't the sequence {x_{n}} not be cauchy? I tried to think of some examples to disprove it but i didnt achieve anything doing that, please...
  29. C

    Proving Cauchy Sequence Subsequences

    I need help on trying to prove that every subsequence of a cauchy sequence is a cauchy sequence
  30. Oxymoron

    Cauchy sequences in an inner product space

    Im in need of some guidance. No answers, just guidance. :smile: Question. Let (x_m) be a Cauchy sequence in an inner product space, show that \left\{\|x_n\|:n=1,\dots,\infty\right\} is bounded. proof From the definition we know that all convergent sequences are Cauchy...
  31. R

    Cauchy integral problem: can this answer be simplified further?

    The question calls for using Cauchy's integral formula to compute the integral for Int.c z/[(z-1)(z-3i)] dz, assuming C is the loop |z-1|=3. Taking z = 1 and f(z) = z/(z-3i), I came up with (2pi*i)/(1-3i), which seems like it could be simplified, but I'm not sure how.
  32. Y

    What is the significance of the Cauchy horizon?

    I am intriqued by a recent series of three papers on black holes: http://www.arxiv.org/abs/gr-qc/0411060 Title: The river model of black holes Authors: Andrew J. S. Hamilton, Jason P. Lisle (JILA, U. Colorado) http://www.arxiv.org/abs/gr-qc/0411061 Title: Inside charged black holes I...
  33. E

    Cauchy Criterion for Series

    I know that if the series of (a)n (n is a subscript) converges, then the lim (a)n=0. How can I show that if the series of (a)n converges, then lim n(a)n=0? Or rather if a1 +a2 +a3 +...+an=0, then lim n*(a)n=0? Not sure how to show this, but I know the proof involves the cauchy criterion...
  34. T

    Prove Cauchy Sequence: {sn} from {tn}

    Let {an}(n goes from 1 to infinity) be a sequence. For each n define: sn=Summation(j=1 to n) of aj tn=Summation(j=1 to n) of the absolute value of aj. Prove that if {tn}(n goes from 1 to infinity) is a Cauchy sequence, then so is {sn}(n goes from 1 to infinity). I started this...
  35. E

    Cauchy Riemann Conditions Question

    Ok, I am told in a complex analysis book that the gradient squared of u is equal to the gradient squared of v which is equal to 0. We know the derivate of w exists, and w(z)=u(x,y) + iv(x,y) Thus the Cauchy Riemann conditions must hold. (When I use d assume that it refers to a partial...
  36. S

    Cauchy Integral Formula and Electrodynamics

    Is it possible to solve for an E field from a charge density function using the Cauchy Integral Formulas from complex variables? Cauchy Integral Formula about a closed loop in the complex plane (Integral[f[z]/ (z-z0)^(n+1)dz = 2 pi i /n! d^n f(z0)/dz ]) that is the n derivative of f with...
  37. I

    Cauchy Mean Value Theorem Proof for Continuous and Integrable Functions

    Hi, I really need some help in sovling this proof! Prove the Cauchy Mean Value Theorem: If f,g : [a,b]->R satisfy f continuous, g integrable and g(x)>=0 for all x then there exists element c is a member of set [a,b] so that int(x=b,a)f(x)g(x)dx=f(c)int(x=b,a)g(x)dx. Thanks for your help :D
  38. B

    Is There Hope for Safe Passage Through Cauchy Horizon Singularities?

    Did anyone else hear about this new development? Apparently if a black hole has a steady influx of matter/energy, it may not develop a singularity which brings about infinite tidal distortion, but it could bring about a 'gentler' cauchy horizon singularity that could be possible to traverse...
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