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. R

    Show That x(n) is a Cauchy Sequence for 0<r<1

    If 0<r<1 and |x(n+1) - x(n)| < r ^n for all n. Show that x(n) is Cauchy sequence. help please.
  2. T

    Analytic Functions Cauchy & Riemann Equations

    Hi, this is fairly fundamental and basic, but I cannot seem to make sense of it I know z = x + iy and hence a function of this variable would be in the form h = f(z). BUT I do not understand why f(z) = u(x,y) + iv(x,y) why so? in z = x + iy, x is the real part and iy is the imaginary...
  3. F

    Cauchy sequence and topological problems

    [PLAIN]http://img805.imageshack.us/img805/1575/photo0138d.jpg Hi, on thursday, i have exam of advance calculus and i could not solve two problem in study sheet given by İnstructor. By 9 question, i prove by add an subtract XnY to |XnYn-XY| and i have found that |Xn(Yn-Y)+...
  4. D

    Complex integral without cauchy

    Hey, I've been trying to do this integral without cauchy's theorem (with the theorem i get 6ipi in like 2 steps). I am stuck at this point, I have found afew ways to do the integral I am stuck on but they all involve multiple variable changes and I was wondering if there is a simple way to do...
  5. S

    Convergence of a Cauchy sequence

    Since I don't know how to use latex I have posed my question in word file. Yours help is greatly appreciated.
  6. B

    Cauchy Reimann Equations Question?

    I'd like to make sure of something. To begin with d/dx will denote partials. My text (Complex Analysis by Steine and Shakarchi) derives the equality df/dx = (1/i) df/dy. To derive this it considers the difference quotient by letting h be real and purely imaginary in another case. But it let's...
  7. M

    Cauchy problem, method of characteristics

    This question is also posted at (with better formatting): http://www.mathhelpforum.com/math-help/f59/cauchy-problem-method-characteristics-187192.html. Solve the following Cauchy problem \displaystyle \frac{1}{2x}u_x + xu u_y + u^2 = 0, subject to \displaystyle u(x,x) = \frac{1}{x^2}, x >...
  8. R

    Cauchy Problem, Same variables in one fraction

    Homework Statement Solve the following Cauchy problem . y' = (3y2 + t2)/ 2ty y(1) = 1 Homework Equations The Attempt at a Solution I know that the equation is not separable or linear from the get-go. I spoke with my professor and he said it is related to the squared...
  9. P

    Understanding Complex Func., Laplace Transforms & Cauchy Riemann

    I am reading a chapter on Complex Functions, Laplace Transforms & Cauchy Riemann (as part of Control theory) And I don't understand how they arrive at a particular part. [ I tried to type it out in tex, but it takes way too much time so uploaded a screenshot to flickr]...
  10. E

    Cauchy Principal Value integral

    Homework Statement Calculate I=P\int^{\infty}_{- \infty} \frac{e^{ikx}}{x} dxHomework Equations I=P\int^{\infty}_{- \infty} \frac{f(x)}{x-x_0} dx = i \pi f(x_0) + 2 \pi i \sum a_{-1}(z_+) The Attempt at a Solution According to Maple the solution is 2i\pi. Now if I try to calculate it using the...
  11. F

    Solve Cauchy Problem for PDE: exp(-x)dz/dx+{y^2}dz/dy=exp(x)yz

    Hi guys tryin to study for a pde exam and cannot solve this question Find a general solution of the equation exp(-x)dz/dx+{/y(squared)}dz/dy=exp(x)yz (ii) Solve the Cauchy problem, i.e. find the integral surface of this equation passing through the curve . y = ex/3 , z = e ...
  12. P

    Cauchy Reimann & Complex functions

    I am a mechanical engineer who hasn't done any mechanical engineering for close to 20 years & hence forgotten all Mechanical Engineering & all Engineering Mathematics. I need to revise on some Engineering Math now - Calculus, Laplace Transforms etc. I have been doing it for a couple of days...
  13. B

    Can Cauchy Sequences be Bounded? Theorem 1.4 in Introduction to Analysis

    Homework Statement Theorem 1.4: Show that every Cauchy sequence is bounded. Homework Equations Theorem 1.2: If a_n is a convergent sequence, then a_n is bounded. Theorem 1.3: a_n is a Cauchy sequence \iff a_n is a convergent sequence. The Attempt at a Solution By Theorem 1.3, a...
  14. ArcanaNoir

    Please review my proof of Cauchy inequality

    Homework Statement I'm actually only concerned here with proving equality. I would like some review of my proof before I crawl back to my professor again with what I think is a valid proof. The Attempt at a Solution Show: \frac{x_1+x_2+...+x_n}{n}=\sqrt[n]{x_1x_2\cdots x_n} \Leftrightarrow...
  15. J

    Cauchy functions problem for Calculus II

    So I've apparently been given an assignment on Cauchy functions (it says here on the title), but I have no idea what that means. Nevertheless, here's my attempt to solve this problem: Given (1): f(x)=\frac{k}{\pi}\times\frac{1}{k^{2}+(x-\beta)^{2}} and (2): \hat{\beta}_{k}=arg\ max_{\beta}...
  16. H

    Using 2nd-Piola-Kirchoff stress rather than Cauchy

    Hi, I need to use the 2nd Piola Kirchoff Stress (S) rather than the usual Cauchy stress i(sigma) in a general transport equation of a material. I was wondering if I need to replace the stress tensor (usually denoted by sigma) with S, or do I just write sigma in terms of S using the relationship...
  17. M

    Cauchy Sequences - Complex Analysis

    Hope someone could give me some help with a couple of problems. First: Proof of - A function f:G -->Complex Plane is continuous on G iff for every sequence C(going from 1 to infinity) of complex numbers in G that has a limit in G we have limit as n --> infinity f(C) = f(limit as n...
  18. G

    Cauchy vs. Dirichelt/Neumann Condition for PDE

    Hi, Can anybody tell me the difference between a Cauchy Boundary condition and a combined Dirichlet/Neumann Boundary Condition for PDEs? The reason why I'm asking is because Cauchy boundary conditions can be used to solve Open Hyperbolic PDEs, whereas Dirichlet/Neumann can only be used to...
  19. Z

    What do the single bars in the Cauchy-Schwarz equation represent?

    I need to write out the proof for the Cauchy-Schwarz equation from quantum computing. I'm stuck on the first step as I don't understand what the single bars on the first term in the equation. Double bars is length but single?? http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality
  20. L

    Cauchy Integral Formula - Multiple Possible Solutions?

    Cauchy Integral Formula -- Multiple Possible Solutions? I'm in the process of teaching myself some complex analysis, and I ran into the following conundrum: Homework Statement Suppose one was given an integral, \int_C \frac{1}{(z-a)(z-b)} \mathrm{d}z where C is a closed curve with a and...
  21. Y

    Sequence of projection is Cauchy

    Homework Statement Let {M_i} be an orthogonal sequence of complete subspaces of a pre-Hilbert space V, and let P_i be the orthogonal projection on M_i. Prove that {P_i(e)} is Cauchy for any e in V 2. The attempt at a solution I'm trying to prove as n and m goes infinity...
  22. R

    What are the roots of the characteristic equation 2m^2 + (1-1)m - 1 = 0?

    Homework Statement Solve: 2x2y" + xy' - y = 3x4 Homework Equations The Attempt at a Solution I know this is a Euler Cauchy solution because of the x2 before the y" in the ODE. I try to solve the characteristic equation m2 +(a-1)m + b = 0 I get 2m2 + (1-1)m - 1 =0 or 2m2 -...
  23. K

    What is the Inverse CDF of Wrapped Cauchy Distribution?

    I'm trying to calculate the inverse CDF of wrapped Cauchy distribution using Mathematica but it gets me nowhere. Probably i lack all the needed knowledge to do so (still a freshman with no statistics experience so far). Any help would be appreciated. Thanks,
  24. D

    Proof of Cauchy Schwarz for complex numbers

    Homework Statement I am told to try and solve <x - ty, x - ty> where t = <x,y>/<y,y> However, I am stuck at that equation and unable to manipulate it to get rid of the * Homework Equations The Attempt at a Solution <x - ty, x - ty> = <x,x> - <x,ty> - <ty,x> + <ty,ty>...
  25. icystrike

    Cauchy Riemann Equation

    Homework Statement What does it mean by this: The cauchy riemann equations are never satisfied when x and y are different from zero and when x=y=0 . Looking at the example of f(z)= l z l = \sqrt{x^{2}+y^{2}} Homework Equations The Attempt at a Solution
  26. S

    Basic Complex Analysis: Cauchy Riemann

    Homework Statement Let f be a holomorphic function in the unit disc D1 whose real part is constant. Prove that the imaginary part is also constant. Homework Equations Cauchy Riemann equations The Attempt at a Solution Hi guys, I'm working through the basics again. I think here we...
  27. J

    Question about Cauchy Theorem to Abelians groups

    Let G group and N subgroup normal from G if b \in{G} and p is prime number then (Nb)^p=Nb^p, Please help me with steps to this proof.
  28. B

    Proving Cauchy Sequence Converges on Real Number Line

    I know about the proof using lim inf and lim sup and the proof using a convergent subsequent, however I thought about this proof. Can you tell me if it is correct, and if not why? Thank you let Sn be Cauchy seq in R Let S be its range. Then S is bounded. Since R is complete, sup...
  29. F

    Proof of the Cauchy Shwarz inequality

    My mathematical methods for theoretical physics course recently began looking at linear vector spaces. We defined the Banach and Hilbert Spaces and proved the Cauchy-Shwarz Inequality. There's one step in this proof that I can't really follow (in red): consider: w=x+uy (i'll drop the...
  30. Fredrik

    What do French, German, and Russian texts call the Cauchy-Schwarz inequality?

    I was once told that the inequality that most books in English seem to call "the Cauchy-Schwarz inequality" is called "Cauchy's inequality" in France, "Schwarz's inequality" in Germany (or Austria or whatever...I'm too lazy to find out where he's from), and "Bunyakovsky's inequality" in Russia...
  31. E

    Cauchy Sequences in General Topological Spaces

    "Cauchy" Sequences in General Topological Spaces Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every...
  32. M

    Solving Cauchy PDEs using the Method of Characteristics

    Could someone tell me where to start? I tried separating variables, which got me no where (plus we haven`t technically learned it), and I tried putting it into a form of D^2U, but I couldn`t figure that out either. Please help. Thank you.
  33. T

    Cauchy Sequence and Completeness

    Homework Statement let (X,d) be a metric space and let A be a dense subset of X such that every Cauchy sequence in A converges in X. Prove that (X,d) is complete. Homework Equations (X,d) is complete if all Cauchy sequences in X converge. A is a dense subset of X => closure(A) = X...
  34. X

    Need example of a continuous function map cauchy sequence to non-cauchy sequence

    Homework Statement I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces. Homework Equations If a function f is continuous in metric space (X, d), then...
  35. M

    Cauchy Product: Exponents or Counters Equal?

    When attemtpting to find the Cauchy product of two functs f(x) and g(x), which are themselves power series, is it more important to have their respective terms of summation be the same of for the exponent of their respective x variable to be equal? Or must both of these conditions be met? I am...
  36. R

    Cauchy riemann equations and constant functions

    Homework Statement Let f(z) be analytic on the set H. Let the modulus of f(z) be constant. Does f need be constant also? Explain. Homework Equations Cauchy riemann equations Hint: Prove If f and f* are both analytic on D, then f is constant. The Attempt at a Solution I think f need be...
  37. M

    Uniform continuity, cauchy sequences

    Homework Statement If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} is also cauchy. Homework Equations The Attempt at a Solution Since f is uniformly continuous, \forall\epsilon>0, \exists\delta>0: \forallx, y ∈ S, |x-y| < \delta =>...
  38. A

    Cauchy Riemann Equations (basic doubt)

    Lets say we have a function of a complex variable z , f(z). I read that for the function to be differentiable at a point z0 , the CR equations are a necessary condition but not a sufficient condition. Can someone give me an example where the CR equations hold but the function is not...
  39. P

    Calculating Integrals with Cauchy Formula

    Homework Statement using cauchy integral formula calculate \int\limits_C\frac{e^{2z}}{z^2-4}\mbox{d}z where C is closed curve (point z=2 is inside) The Attempt at a Solution \ldots=\int\limits_C\frac{e^{2z}}{(z-2)(z+2)}\mbox{d}z=\int\limits_C\frac{f(z)}{z-2}\mbox{d}z=2\pi if(2)=2\pi...
  40. M

    Proving Cauchy Sequences Using the Definition

    Homework Statement Im trying to prove that a sequence is cauchy using its definition. the sequence is 81+14n/50+31n. i know it converges to 14/31 using L'hopitals rule but the assignment is to use the definition of cauchy i have tried some things but I am not 100% sure how to start all...
  41. A

    Cauchy Convergence in Normed Vector Spaces

    Homework Statement Fix a<b in R, and consider the two norms Norm(f)1:=Integralab( Modulus(f)) and Norm(f)Infinity:= sup{Mod(f(x)): a <= x <= b} on the vector space C[a,b]. This question shows that they are not equivalent. a. Show that there is K in R such that for all f in C[a,b]...
  42. B

    Proving Cauchy Sequence with Triangle Inequality

    Homework Statement Let Sn be a sequence such that |Sn+1-Sn|< 2-n for all n in the natural numbers Homework Equations The Attempt at a Solution I understand what it means to be cauchy but I'm not sure how to prove this particular sequence is cauchy. Please help!
  43. K

    Is Every Convergent Sequence in a Closed Set a Cauchy Sequence?

    Homework Statement A set F\subseteqR is closed iff every Cauchy sequence contained in F has a limit that is also an element of F. Homework Equations The Attempt at a Solution Let F be closed. Then F contains its limit points. This means x=lima_{n} are elements of F.
  44. D

    Cauchy/ Triangle Inequality

    I was wonder what conditions a and b have to be for each inequality in order to satifsy the equality?
  45. J

    Bolzano-wierstrass theorem and its dilemma when applied to cauchy sequence

    From this theorem, we know that every cauchy sequence is bounded so there exists an accumulation point a\in \mathbb{R}^k. This is the limit point of the cauchy seqence. It shows that there is only one accumulation point as limit point is unique. However is it a must that there is only one...
  46. H

    Showing that there's a Cauchy sequence where Xn<X for all n?

    Homework Statement "If x is a real number, show that there exists a Cauchy sequence of rationals Xl, X2,... representing X such that X n < x for all n." Homework Equations - All Cauchy sequences are convergent - All Cauchy sequences are bounded. The Attempt at a Solution These proofs...
  47. J

    How to Use Cauchy Integral Formula for Homework Problems?

    Homework Statement 1) Evaluate \intc ((5z-2)/(z(z-1)(z-3)))dz where c is the circle of radius 2 about the origin. 2) Evaluate \intc (2*(z^2)-z+1) / ((z-1)^2(z+1))dz where c proceeds around the boundary of the figure eight formed by two circles of radius 1 with centres 1 and -1 by starting at...
  48. ╔(σ_σ)╝

    Show that the Harmonic series is not cauchy

    Homework Statement Let x_n = \sum_{k=1}^{n}\frac{1}{k} Show x_n is not cauchy. It seems like a fairly easy problem . I bet my head is just not in the right place tonight ( It's thanksgiving in Canada :D) .Homework Equations The Attempt at a Solution Well I know it is not bounded hence...
  49. G

    Propagator using cauchy integral

    hi I don't understand this bit about the derivation of propagator expressions. Bjorken and Drell describe the step function as: \theta(\tau)=lim_{\epsilon \to 0}\frac{-1}{2\pi i}\oint_{-\infty}^{\infty}\frac{d\omega e^{-i\omega r}}{\omega + i \epsilon } the singularity is at -i \omega...
  50. J

    Cauchy Sequence Convergence in the Real Numbers

    hello, i have 2 gre questions: 1) if i have a sequence xn in R that goes to 0, then: a) if f is a continuous function , then f(xn) is a cauchy sequence. (true ?) b) if f is a uniformly continuous function , then f(xn) is a convergent sequence. (true?) 2) lim z--> 0...
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