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. D

    Can the Cauchy Product of Series be Derived by Redefining Dummy Variables?

    Hey guys, I was just doing some independent study on products of series and I'm trying to understand/derive the following form of the Cauchy product of series: \left(\sum_{n=0}^{N} a_{n}\right) \left(\sum_{m=0}^{N} b_{m}\right) = \sum_{n=0}^{N} \left(\sum_{k=0}^{n} a_{k}b_{n-k}\right)...
  2. A

    MHB What is the Cauchy Integral Theorem and How Does it Apply to Complex Numbers?

    Show that the whole complex has zero following result (Cauchy Integral Theorem):
  3. O

    MHB Why is a boundary condition at x=0 redundant for this Cauchy problem?

    I have this cauchy problem U_t(x,t)= c_0[tanhx]u_x(x,t)=0 U(x,0)= u_0(x) I managed to prove that it has at most one solution my question is why would it be redundant to have a boundary condition at x=0
  4. A

    MHB How to use cauchy integral formula

    Hello. How do I know when to use Cauchy integral formula. Why do we use the formula in this question? As you can see in my attempt, I got stuck. What is f(z), z, z​0 here?
  5. S

    MHB Proving Cauchy Convergence in Natural Nos: Metrics & Converse

    Using the fact that the Natural Nos are complete .then prove that every Cauchy sequence in Natural Nos converges in N and the converse. I do not even know if we can have a Cauchy sequence in Natural Nos. What would be the appropriate metric to use in our Cauchy sequence??
  6. Z

    What is the meaning and purpose of the Cauchy Principal Value in integrals?

    Hello everyone, I have recently bumped into the Kramers Kronig Relations while reviewing some of my Eletromagnetism notes, and as you may know those relations are written in terms of the Cauchy Principal Value (CPV) of certain integrals. Well, I've never been very familiar with with the...
  7. A

    MHB An Equivalence Relation with Cauchy Sequences

    We let C be the set of Cauchy sequences in \mathbb{Q} and define a relation \sim on C by (x_i) \sim (y_i) if and only if \lim_{n\to \infty}|x_n - y_n| = 0. Show that \sim is an equivalence relation on C. We were given a hint to use subsequences, but I don't think they are really necessary...
  8. J

    Why is ε→0⁺ used in both terms of the Cauchy principal value formula?

    The cauchy principal value formula is: But why ε→0⁺ in both terms? The correct wouldn't be ε→0⁻ in 1st term and ε→0⁺ in 2nd term? Like: \lim_{\varepsilon \to 0^-}\int_{a}^{c-\varepsilon}f(x)dx + \lim_{\varepsilon \to 0^+}\int_{c+\varepsilon}^{b}f(x)dx ?
  9. D

    Integral using cauchy theorem

    Hello, I don't get why using the fact that ∫dz/z = 2*pi*i accros the circle This integral gives: 1/3∫(1/(z-2)-1/(z-1/2))dz = 1/3(-2*pi*i) across the circle Thanks !
  10. B

    Linear algebra 1: cauchy schwarz problem

    Homework Statement If llull = 4, llvll = 5 and u dot v = 10, find llu+vll. u and v are vectors Homework Equations llu+vll = llull + llvll cauchy schwarz The Attempt at a Solution (1) llu+vll = llull + llvll (2) (llu+vll)^2 = (llull + llvll)^2 (3) (llu+vll)^2 = llull^2 +...
  11. D

    Cauchy Sequence Homework: Show x_n is Cauchy

    Homework Statement Given: x_{n+1}=\frac{1}{3+x_n} with x_1=1 Show that: (1) |x_{n+1}-x_n| \leq \frac{1}{9}|x_{n}-x_{n-1}| and (2) x_n is Cauchy. Homework Equations The Attempt at a Solution I've tried different approaches (including induction) but the...
  12. alyafey22

    MHB Complete spaces and Cauchy sequences

    I know that a metric space is complete if every Cauchy sequence converges that will surely designate compact metric spaces as complete spaces . I need to see examples of metric spaces which are not complete. Thanks in advance !
  13. P

    Help with Cauchy Integral Formula

    Use Cauchy Integral Formula to solve: ∫ [(5z² - 3z + 2)/(z-1)³] dz C is any closed simple curve involving z=1. (z is a complex) Thanks and sorry for my poor english, it's not my first language.
  14. U

    Two dimensional Cauchy problems

    Let y(t) = (y1(t), y2(t))^T and A(t) = (a(t) b(t) c(t) d(t)). A(t) is a 2x2 matrix with a,b,c,d all polynomials in t. Consider the two dimensional Cauchy problem y'(t) = A(t)y(t), y(0)=y0. Show that a solution exists for all t>=0. Give a general condition on the A(t) which ensures...
  15. C

    Complex integration (Using Cauchy Integral formula)

    Homework Statement $$\int_\gamma \frac{\cosh z}{2 \ln 2-z} dz$$ with ##\gamma## defined as: 1. ##|z|=1## 2. ##|z|=2## I need to solve this using Cauchy integral formula. Homework Equations Cauchy Integral Formula The Attempt at a Solution With ##|z|=2## I've solved already, as it is...
  16. X

    Example of cauchy sequence

    one of example of cauchy sequence show that = 1/n - 1/(n+k) and In the above we have used the inequality 1/(n+m)^2 <= ( 1/(n+m-1) - 1/(n+m) ) => i don't under stand where this come from and what is inequality? can you give other example?
  17. M

    Understanding the Cauchy Stress Tensor: A Guide for Beginners

    Hello, I am not sure what the first indice in the cauchy stress tensor indicates For example, σ_xy means that the stress in the y direction, but does x mean the cross sectional area is normal to the x direction?
  18. M

    Calculating Contour Integrals with Cauchy Theorem on Annulus/Donut Boundaries

    A complex analysis question. Homework Statement Verify the Cauchy theorem by calculating the contour integrals. Where ω is the appropriately orientated boundary of the annulus/donut defined by 1/3 ≤ IzI ≤ 2 for the following analytic functions: i. f(z)=z^2 ii. f(z)=1/z Homework...
  19. D

    Cauchy Schwarz proof with alternative dot product definition

    Homework Statement Does the Cauchy Schwarz inequality hold if we define the dot product of two vectors A,B \in V_n by \sum_{k=1}^n |a_ib_i| ? If so, prove it. Homework Equations The Cauchy-Schwarz inequality: (A\cdot B)^2 \leq (A\cdot A)(B\cdot B) . Equality holds iff one of the vectors...
  20. E

    Cauchy expansion of determinant of a bordered matrix

    The Cauchy expansion says that \text{det} \begin{bmatrix} A & x \\[0.3em] y^T & a \end{bmatrix} = a \text{det}(A) - y^T \text{adj}(A) x , where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar. There is a proof in Matrix Analysis by Horn and...
  21. A

    The difference between the limits of two Cauchy Sequences

    Lets say that we have two Cauchy sequences {fi} and {gi} such that the sequence {fi} converges to a limit F and the sequence {gi} converges to a limit G. Then it can easily be shown that the sequence defined by { d(fi, gi) } is also Cauchy. But is it true that this sequence, { d(fi, gi) }...
  22. A

    Cauchy sequence problem

    Homework Statement Let (M,d) be a complete metric space and define a sequence of non empty sets F1\supseteqF2\supseteqF3\supseteq such that diam(Fn)->0, where diam(Fn)=sup(d(x,y),x,y\inFn). Show that there \bigcapn=1∞Fn is nonempty (contains one element). Homework Equations The...
  23. P

    Proving a sequence is a cauchy sequence in for the 7 -adic metric

    Homework Statement Show that the sequence (xn)n\inN \inZ given by xn = Ʃ from k=0 to n (7n) for all n \in N is a cauchy sequence for the 7 adic metric. Homework Equations In a metric space (X,dx) a sequence (xn)n\inN in X is a cauchy sequence if for all ε> 0 there exists some M\inN such...
  24. G

    Cauchy sequence and convergeant diameters.

    Suppose (an) is sequence in the metric space X and define Tn={ak:k>n} and diamT=sup{d(a,b):a,b elements of T}. Prove that (an) is Cauchy if and only if diam Tn converges to zero. In what metric spacee does Tn converge? I assumed in (ℝ,de) but this is confusing since the diam of T is...
  25. A

    Solving Cauchy Residual Theorem for Holomorphic Function at z=2i

    Alright so I posted a picture asking the exact question. Here is my best attempt... According to my professor's terrible notes, the numerator can magically turn into the form: e^i(z+3) when converted to complex. The denominator will be factored into (z-2i)(z+2i) but the...
  26. B

    Alternative Proof of Cauchy Sequence ##\left(S_n\right) = \frac{1}{n}##

    I am looking for a different proof that ##(S_n) = \frac{1}{n}## is cauchy. The regular proof goes like this (concisely): ##\left|\frac{1}{n} - \frac{1}{m} \right| \leqslant \left|\frac{m}{nm}\right| \ (etc...) \ <\epsilon ## but I was thinking about an alternative proof. Is my proof...
  27. J

    Cauchy schwarz inequality in Rudin

    I have worked my way though the proof of the Cauchy Schwarz inequality in Rudin but I am struggling to understand how one could have arrived at that proof in the first place. The essence of the proof is that this sum: ##\sum |B a_j - C b_j|^2## is shown to be equivalent to the following...
  28. S

    What units should be used for the cauchy dispersion formula?

    Hi everybody, I would like to use the 'cauchy dispersion formula', ie (http://en.wikipedia.org/wiki/Cauchy's_equation"]http://en.wikipedia.org/wiki/Cauchy's_equation):[/PLAIN] eta = A + B / w² Where : eta is the resulting IOR A is the base IOR B is the dispersion coefficient expressed...
  29. alyafey22

    MHB Proof of Cauchy Integral Formula

    I want to prove the Cauchy integral formula : \oint_{\gamma} \, \frac{f(z)}{z-z_0}\,dz= 2\pi i f(z_0) \text{so we will integrate along a circle that contains the pole .} |z-z_0|= \delta \,\text{ which is a circle centered at the pole and has a radius }\delta\,\, z =z_0+\delta e^{i\theta...
  30. S

    Calculating Residues and using Cauchy Integral Formula

    Hey, I have a problem with this integral: \int_{-\infty}^{\infty}dE\frac{1}{E^{2}-\mathbf{p}^{2}-m^{2}+i\epsilon}\: ,\: l^{2}=\mathbf{p}^{2}+m^{2} The integration over all energies (arising in the loop function for calculating the scattering), I understand we write the above in this form...
  31. R

    Complex Analysis - Cauchy Integral? Which technique do I use?

    Homework Statement \int_0^\infty\frac{x^{p-1}}{1+ x}dx ** I could not get p-1 to show as the exponent; the problem is x raised to the power of p-1. \int_0^\infty\frac{ln(x) dx}{(x^2+1)^2} The Attempt at a Solution There is no attempt, but I would like to make one! I'm asking...
  32. S

    What does the N mean in a Cauchy sequence definition?

    What does the "N" mean in a Cauchy sequence definition? Hi everyone, I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question. I believe I have an intuitive understanding of what a Cauchy...
  33. P

    Cauchy Problem in Convex Neighborhood

    While reading the reference Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html, there is something that I don't quite understand. Eq.(16.6) is an evolution equation for...
  34. H

    Determining if a sequence is convergent and/or a Cauchy sequence

    Homework Statement Let {pn}n\inP be a sequence such that pn is the decimal expansion of \sqrt{2} truncated after the nth decimal place. a) When we're working in the rationals is the sequence convergent and is it a Cauchy sequence? b) When we're working in the reals is the sequence...
  35. S

    Cauchy Sequences and Convergence

    Homework Statement Prove the following theorem, originally due to Cauchy. Suppose that (a_{n})\rightarrow a. Then the sequence (b_{n}) defined by b_{n}=\frac{(a_{1}+a_{2}+...+a_{n})}{n} is convergent and (b_{n})\rightarrow a. Homework Equations A sequence (a_{n}) has the Cauchy property...
  36. P

    Any Cauchy sequence converges.

    Hello, My instructor, whilst trying to prove that liminf of sequence a_n = limsup of sequence a_n = A, _ wrote that since we know that a_n0-ε<an<a_n0+ε → a_n0-ε ≤ A ≤ A ≤ a_n0+ε...
  37. F

    Solving Trigonometric integrals using cauchy residue theorem

    Homework Statement evaluate the given trigonometric integral ∫1/(cos(θ)+2sin(θ)+3) dθ where the lower limit is 0 and the upper limit is 2π Homework Equations z = e^(iθ) cosθ = (z+(z)^-1)/2 sinθ = (z-(z)^-1)/2i dθ = dz/iz The Attempt at a Solution after I substitute and...
  38. M

    Using Cauchy integral formula to compute real integral?

    Homework Statement Compute the following integral around the path S using Cauchys integral formula for derivatives: \intez / z2 Integral path S is a basic circle around origin. Then, use the result to compute the following integral \int ecos (x) cos(sin (x) - x) dx from 0 to ∏...
  39. S

    Proof of Cauchy integral formula

    Homework Statement For an quiz for a diff eq class, I need to prove the Cauchy integral formula. The assignment says prove the formula for analytic functions. Is the proof significantly different when the function is not analytic? Homework Equations Basically, a proof I found online says...
  40. S

    Cauchy sequences is my proof correct?

    Homework Statement Let (xn)n\inℕ and (yn)n\inℕ be Cauchy sequences of real numbers. Show, without using the Cauchy Criterion, that if zn=xn+yn, then (zn)n\inℕ is a Cauchy sequence of real numbers. Homework Equations The Attempt at a Solution Here's my attempt at a proof: Let...
  41. M

    Prove: Cauchy sequences are converging sequences

    Homework Statement I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence Homework Equations What I know is: a[n] is bounded any subsequence is bounded there exists a monotone subsequence all monotone bounded sequences converge there exists a...
  42. F

    Proof of "Every Cauchy Sequence is Bounded

    I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded I don't understand that how we can take m=N_{1} while m>N_{1} ? In fact i mean that in a metric space (A,d) can we say that...
  43. K

    Proving cauchy criterion for limits

    Homework Statement Prove the converse of the Cauchy Criterion for Limits. Let I be an interval that either contains the point c or has c as one of its endpoints and suppose that f is a function that is defined on I except possibly at the point c. Then the function f has limit at c iff for...
  44. T

    Cauchy Riemann conditions for analyticity for all values of z.

    Homework Statement Show that sin(z) satisfies the condition. (Stated in the title) Homework Equations The Attempt at a Solution f(z) = sin (z) = sin (x + iy) = sin x cosh y + i cos x sinh y thus, u(x,y)=sin x cosh y ... v(x,y)= cos x sinh y du/dx = cos x...
  45. M

    What is the role of the principal value in the Cauchy principal value integral?

    Hi, I came across this for the first time today. \int_0^\infty e^{i\omega t}dt = \pi\delta(\omega)+iP(\frac{1}{\omega}) Here P(\frac{1}{\omega}) is the so called principal value. I haven't seen this term normally so can I ask where we get it from? Googling principal value showed me a very...
  46. N

    Integral for (kind of) standard Cauchy distibution and an alternative solution.

    Hello! I have a couple of questions on the following. Firstly, I was hoping someone could check my working and my reasoning. Secondly, I was wondering if someone knew an alternative way of solving this problem. I wanted to integrate this from x = -\infty to x = \infty: \lim_{a...
  47. D

    Use Cauchy Residue Theorem to find the integral

    Homework Statement To find the integral by Cauchy Residue Theorem and apply substitution method. Homework Equations To show: ∫^{2∏}_{0}\frac{cosθ}{13+12cosθ}=-\frac{4∏}{15} The Attempt at a Solution The solution I have done is attached. It is different as what the question wants me...
  48. fluidistic

    Cauchy-Schwarz Inequality Proof | MathWorld Demonstration and Solution

    Homework Statement I'm trying to follow the demonstration of the Cauchy-Schwarz's inequality proof given in http://mathworld.wolfram.com/SchwarzsInequality.html. I am stuck at the last step, namely that \langle \bar g , f \rangle \langle f , \bar g \rangle \leq \langle \bar f , f \rangle...
  49. D

    Use Cauchy Integral Formula to evaluate the integral

    Homework Statement The question is needed to be done by using an appropriate substitution and the Cauchy Integral Formula. Homework Equations Evaluate the complex integral: ∫e^(e^it) dt, from 0 to 2∏ The Attempt at a Solution I cannot find an appropriate substitution for the integral.
  50. E

    How to determine particular solutions for cauchy euler

    If given a cauchy euler equation (non-homogeneous) equation, does the approach in looking for a particular solution (in order to solve the non-homogeneous part), differ from normal? I am also in general confused about how to assign a particular solution form, in many cases. I have yet to find...
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