Cauchy Definition and 381 Threads

Homework Statement I need to solve: x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9 Homework EquationsThe Attempt at a Solution I know that the answer is: y=x^2+2x^3+x^3lnx Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...

If $\{x_n\}_{n \ge 1}$ is real sequence and $\limsup\limits_{n \to \infty} \frac{1}{n} \log |x_{n+1}-x_n|<0$, prove that $\{x_n\}$ is Cauchy sequence. My work: Let $a=\limsup\limits_{n \to \infty} \frac{1}{n} \log |x_{n+1}-x_n| <0$. Then, for every $\varepsilon >0$ there exist $N \in...

Homework Statement https://en.wikipedia.org/wiki/Cauchy_stress_tensor[/B] I don't understand the difference between τxy . τyx , τxz , τzx , τyz , τzy ..What did they mean ? Homework EquationsThe Attempt at a Solution taking τxy and τyx as example , what are the difference between them ? They...

From "Cauchy Momentum Equation" on Wikipedia, The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes Fi This is exactly what I am having trouble grasping. It's probably something simple and...

Hi, I have a question about Cauchy Reimann equation lets say z=x+yi is in R^2 And there exists f:R^2->R^2 f(z)=u(x,y)+v(x,y)i Then cauchy reimann condition states that If partial x of f and y of f are equal, then f is holomorphic However, I am not sure how this can be a necesary sufficient...

I have a set of data which are probably convolutions of a Cauchy distribution with some other distribution. I am looking for some model for this other distribution so that a tractable analytic formula results. I know that the convolution Cauchy with Cauchy is again Cauchy, but I want the other...

Homework Statement Find the residues of the function f(z), and compute the following contour integrals. a) the anticlockwise circle, centred at z = 0, of radius three, |z| = 3 b) the anticlockwise circle, centred at z = 0, of radius 1/2, |z| = 1/2 f(z) = 1/((z2 + 4)(z + 1)) ∫Cdz f(z) Homework...

Homework Statement So I've found a ton of examples that show you how to find cauchy principal values of convergent integrals because it is just equal to the value of that integral and you prove that the semi-circle contribution goes to zero. However, I need to find some Cauchy principal values...

https://anhngq.wordpress.com/2013/04/25/the-cauchy-formula-for-repeated-integration/ Could someone please explain, bit by bit, how the formula works? For instance, why are we integrating with respect to sigma, up to sigma in the equation before it?

Find $ \oint\frac{e^{iz}}{z^3}dz $ where contour is a square, center 0, sides > 1 There is an interior pole of order 3 at z=0 CIF: $ \oint\frac{f(z)}{(z-z_0)^{n+1}}dz = \frac{2\pi i}{n!} f^{(n)}(z_0) = \frac{2 \pi i}{2}f''(z_0) = -\pi i $

Problem A now solved! Problem B: I am working with two equations: The first gives me the coefficients for the Laurent Series expansion of a complex function, which is: f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n This first equation for the coefficients is: a_n = \frac{1}{2πi} \oint...

Hello evry body let be $(u_{n}) \in \mathbb{C}^{\matbb{N}}$ with $u_{n}^{2} \rightarrow 1$ and $\forall n \in \mathbb{N} (u_{n+1) - u_{n}) < 1$. Why does this sequences converge please? Thank you in advance and have a nice afternoon:oldbiggrin:.

Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...

Let $\left\{{x}_{n}\right\}$ be a sequence...İf $\left\{{x}_{2n}\right\}$ is caucy sequence, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence ?

I'm learning about Support Vector Machines and would like to recap on some basic linear algebra. More specifically, I'm trying to prove the following, which I'm pretty sure is true: Let ##v1## and ##v2## be two vectors in an inner product space over ##\mathbb{C}##. Suppose that ## \langle v1 ...

A Cauchy surface is a 3d spacelike slice of spacetime. Could you read the definition (17) p18 the author says that "one can proove that ##\sigma## does not depend on the choice of the Cauchy surface because the functions obey the law of motion." Could you elaborate? Thanks Could anyone add "h"...

Homework Statement \int_{0}^{2\pi} \dfrac{d\theta}{3+tan^2\theta} Homework Equations \oint_C f(z) = 2\pi i \cdot R R(z_{0}) = \lim_{z\to z_{0}}(z-z_{0})f(z) The Attempt at a Solution I did a similar example that had the form \int_{0}^{2\pi} \dfrac{d\theta}{5+4cos\theta} where I would change...

My Questions: 1) İn both sides of inequality of (*) why we use "n", that is, why we do multiplication with "n" ? 2) in (**) by Letting $n\to\infty$ we obtain $\lim_{{n}\to{\infty}} n\left[d\left({T}^{n}x,{T}^{n+1}x\right)\right]{}^{r}=0$ How this...

Find surface of $\begin{array}{l} \text{Problem Cauchy} \\ {a^2} \cdot {x_2} \cdot u \cdot {u_{{x_1}}} + {b^2} \cdot {x_1} \cdot u \cdot {u_{{x_2}}} = 2{c^2}{x_1}{x_2}{\rm{ }} \\ \end{array}$ The partial differntial equation passes through ${\rm{ C: = \{ }}\frac{{{x^2}}}{{{a^2}}} +...

Solve the Cauchy problem: \begin{align} \dfrac{dx}{dr} &= y\\ x(0,s) &= s \end{align} Help please, I don't remember how solve this :(. Separate variable isn't the method?

Homework Statement Suppose the sequence (xn) satisfies |xn + 1 - xn| < 1/n2, prove that (xn) is convergent. Homework Equations |xn - xm| < ɛ The Attempt at a Solution If m > n, then |xn - xm| < |xn - xn + 1| + |xn + 1 - xn + 2| + ... + |xm - 1 - xm| < 1/n2 + 1/(n+1)2 + ... + 1/(m - 1)2 <...

Homework Statement Observe a Cauchy problem \begin{cases}y' + p(x)y =q(x)y^n\\ y(x_0) = y_0\end{cases} Assume ##p(x), q(x)## are continuous for some ##(a,b)\subseteq\mathbb{R}## Verify the equation has a solution and determine the condition for there to be exactly one solution. Homework...

İn some articles, I see something... For example, Let we define a sequence by ${x}_{n}=f{x}_{n}={f}^{n}{x}_{0}$$\left\{{x}_{n}\right\}$. To show that $\left\{{x}_{n}\right\}$ is Cauchy sequence, we suppose that $\left\{{x}_{n}\right\}$ is not a Cauchy sequence...For this reason, there exists a...

Let $\left(E,d\right)$ be a complete metric space, and $T,S:E\to E$ two mappings such that for all $x,y\in E$, $d\left(Tx,Sy\right)\le M\left(x,y\right)-\varphi\left(M\left(x,y\right)\right)$, where $\varphi:[0,\infty)\to [0,\infty)$ is a lower semicontinuous function with...

So the formula for an ellipse in polar coordinates is r(θ) = p/(1+εcos(θ)). By evaluating L = ∫r(θ) dθ on the complex plane on a circle of circumference ε on the centered at the origin I obtained the equation L = (2π)/√(1-ε^2). Why then does Wikipedia say that the formula for the perimeter is...

Hi, looking at a proof of Cauchy Integral formula, I have (at least) one question, starting from the step below $ \int_{{C}}^{}\frac{f(z)}{z-{z}_{0}} \,dz - \int_{{C}_{2}}\frac{f(z)}{z-{z}_{0}} \,dz = 0 $ , where $ {C}_{2}$ is the smaller path around the singularity at $ {z}_{0} $ Let...

I have been working on a math problem and I keep getting the some type of PDEs. x*dU/dx+y*dU/dy = 0 x*dU/dx+y*dU/dy+z*dU/dz = 0 ... x1*dU/dx1+x2*dU/dx2+x3*dU/dx3 + ... + xn*dU/dxn= 0 dU/dxi is the partial derivative with respect to the ith variable. Does anyone know about this type of PDE...

to prove that $\left\{{x}_{n}\right\}$ is Cauchy seqeunce we use a method. I have some troubles related to this method. Please help me... $\left\{{c}_{n}\right\}$=sup$\left\{d\left({x}_{j},{x}_{k}\right):j,k>n\right\}$.Then $\left\{{c}_{n}\right\}$ is decreasing. If ${c}_{n}$ goes to 0 as n...

(Wish there was a solutions manual...). Please check my workings below Show $ \int \frac{dz}{{z}^{2} + z} = 0 $ by separating integrand into partial fractions and applying Cauchy's Integral theorem for multiply connected regions. For 2 paths (i) |z| = R > 1 (ii) A square with corners $ \pm 2...

Hi - I know the final result for the n'th derivative, I am looking though at getting an expression for the 1st derivative of f(z). From $ f({z}_{0}) = \frac{1}{2\pi i} \oint_{c} \frac{f(z)}{z - {z}_{0}}dz $ we get $ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =...

Homework Statement Verify that each of the following functions is entire: f(z)=(z^2-2)e^(-x)e^(-iy) Homework Equations The Cauchy Riemann equations u(x,y) = ______ and v(x,y) = ______ u_y=-v_x u_x=v_y The Attempt at a Solution So, I've done a few of these problems and understand that to...

http://www.math.hawaii.edu/~williamdemeo/Analysis-href.pdf Please look at problem 2 on page 39 of the problems/solutions linked above. I know I'm going to kick myself when someone explains this to me but how was equation "(31)" of the solution obtained? The first term of the RHS of (31) is...

Dear all, I'm not a mathematician so please excuse me for a certain lack of strictness ... I work on random signals in physics, these signals are most of the time called "noise" for us. For example, we can speak about x(t), a time domain random signal. Very usually, the statistics...

Homework Statement I have to prove that I(a,b)=\int_{-\infty}^{+\infty} exp(-ax^2+bx)dx=\sqrt{\frac{\pi}{a}}exp(b^2/4a) where a,b\in\mathbb{C}. I have already shown that I(a,0)=\sqrt{\frac{\pi}{a}}. Now I am supposed to find a relation between I(a,0) and \int_{-\infty}^{+\infty}...

I need to construct a cauchy sequence ${r}_{n}$ such that its rational for all n belonging to set of natural numbers, but the limit of the sequence is not rational when n tends to infinity. I know that all convergent sequences are cauchy but I can't randomly conjure up a sequence that satisfies...

Here is part of the proof via wikipedia: "We first prove the special case that where G is abelian, and then the general case; both proofs are by induction on n = |G|, and have as starting case n = p which is trivial because any non-identity element now has order p. Suppose first that G is...

Hi, I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this: Find all the complex numbers z∈ C such that f(z)=z cos (z ̅). [z ̅ is z-bar, the complex conjugate). Thanks!

I need some help in fully understanding Example 1, section 4.3 Cauchy Sequences, page 73 of Apostol, Mathematical Analysis. Example 1, page 73 reads as follows: https://www.physicsforums.com/attachments/3844 https://www.physicsforums.com/attachments/3845 In the above text, Apostol writes: "...

Hey! (Wave) Sentence: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. Proof: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show that, without loss of generality, we can suppose that $x_i \in \mathbb{Z}_p$.Let...

prove that limit of a cuachy sequence of integers is an integer

I've been very confused with this proof, because if a sequence { 1, 1, 1, 1, ...} is convergent and bounded by 1, would this be considered to be a Cauchy sequence? I'm wondering if this has an accumulation point as well, by using the Bolzanno-Weirstrauss theorem. I really appreciate the help...

Hi! (Wave) I am looking at the following exercise: If $\{ x_n \}$ is a sequence of rationals, then this is a Cauchy sequence as for the p-norm, $| \cdot |_p$, if and only if : $$\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$$ That's what I have tried: $\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$ means...

Homework Statement I want to prove that if X is a normed space, the following statements are equivalent. (a) Every Cauchy sequence in X is convergent. (b) Every absolutely convergent series in X is convergent. I'm having difficulties with the implication (b) ⇒ (a). Homework Equations Only...

A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default? When trying to think of a space with no cauchy...

Hi All, I was reading through Kreyzeig's Advanced Engineering Mathematics and came across two theorems in Complex Analysis. Theorem 1: Let f(z) = u(x,y) + iv(x,y) be defined and continuous in some neighborhood of a point z = x+iy and differentiable at z itself. Then, at that point, the...

Dear friends, I have been told that if ##\{a_n\}_{n\in\mathbb{N}}##, ##\{a_{-n}\}_{n\in\mathbb{N}^+}##, ##\{b_n\}_{n\in\mathbb{N}}## and ##\{b_{-n}\}_{n\in\mathbb{N}^+}## are absolutely summable complex sequences -maybe even if only one i between ##\{a_n\}_{n\in\mathbb{Z}}## and...

Homework Statement I found the function V, which is the conjugate harmonic function for U(x,y)=sin(x)cosh(y). I am attaching my work. It turns out to be a two-term function with trig factors. I am then to write F(Z) in terms of Z, but is plugging in x, and y, in terms of Z into my trig...

http://www.math.uiuc.edu/documenta/vol-ismp/40_lemarechal-claude.pdf I don't understand why we use theta for equation (1) Θ>0 but why α=-θX? Thanks.

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)...

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