Cauchy Definition and 381 Threads

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

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...

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 >...

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...

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]...

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...

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 ...

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...

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...

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...

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}...

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...

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...

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...

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

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...

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...

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 -...

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,

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>...

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

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...

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.

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...

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...

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...

"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...

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.

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...

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...

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...

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...

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 =>...

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...

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...

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...

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]...

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!

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.

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...