# Search results for query: *

1. ### I On error estimates of approximate solutions

The problem is that the Lipschitz constant L must satisfy L \geq \sup\left\{\left|\frac{f(x) - f(y)}{x - y}\right| : (x,y) \in \Omega^2, x \neq y\right\}. (If f is differentiable, then we have \lim_{x\to y} \left| \frac{f(x) - f(y)}{x - y}\right| = |f'(y)| and we would also require L \geq \sup...
2. ### Difference between -3² and (-3)² ?

Depending on the calculator, it might be that the sequence of keys  [-] [x2] is interpreted as (-3)^2 not -(3^2).
3. ### I Verifying properties of Green's function

\tau is regarded as a parameter of the IVP (2,3), so d/dt here means \partial/\partial t. We know that both \bar{E} and 0 are n - 1 times differentiable with respect to t: 0 trivially, and \bar E because it is the solution of the IVP (2,3), so its first n - 1 derivatives with respect to t exist...
4. ### A Gamma function, Bessel function

You can't put \alpha = -|m| for m \in \mathbb{Z} directly into the definition J_\alpha(x) = \sum_{n=0}^\infty \frac{(-1)^n}{n!\Gamma(n + \alpha + 1)}\left(\frac{x}{2}\right)^{2n+\alpha} but that doesn't mean that J_{-m} is not defined. Frobenius's Method for the Bessel equation xy'' +xy'...
5. ### A Gamma function, Bessel function

There is only one complex infinity: It is the single point you must add to the complex plane to make it toplogically conjugate to a sphere. In the real line we treat +\infty and -\infty as being distinct because the real line is ordered. The complex plane is not.
6. ### Taylor Expansion for very small and very big arguments

Use the binomial expansion: (1 + x)^{\alpha} = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^2 + \dots + \frac{\alpha(\alpha - 1) \cdots (\alpha - n + 1)}{n!}x^n + \dots,\qquad |x| < 1. For small |x|, f(x) = (1 - x)^{1/2} can be expanded as is. A Taylor series of x^{1/2} about 1 should also...
7. ### Constraint force using Lagrangian Multipliers

I think you should include the constraint explicitly in the Lagrangian: \mathcal{L} = \tfrac12m(\dot s^2 + s^2 \dot \varphi^2) - mgs\sin \varphi + m\lambda(\varphi - \omega t). How do you justify the last line? You correctly found the derivatives of the Lagrangian, but you didn't put them...
8. ### Find the smallest value of angle ##α + β ##

\frac{\frac{a}{a+1} + \frac{1}{2a+1}}{1 - \frac{a}{a+1}\frac{1}{2a+1}} = \frac{\frac{a}{a+1} + \frac{1}{2a+1}}{1 - \frac{a}{a+1}\frac{1}{2a+1}} \cdot \frac{(a+1)(2a+1)}{(a+1)(2a+1)} = \frac{a(2a+1) + (a+1)}{(2a+1)(a+1) - a} = \frac{ 2a^2 + 2a + 1}{2a^2 + 2a + 1} = 1.
9. ### I Lipschitz continuity of vector-valued function

To guarantee boundedness of the derivatives we require that they be continuous (hence C^1) on a compact domain. This is the generalisation of the theorem that a function continuous on a closed, bounded interval is bounded, but a function continuous on an unclosed bounded interval may not be...
10. ### I Lipschitz continuity of vector-valued function

A basic upper bound for the value of an integral of F: [a,b] \to \mathbb{R} is \int_a^b F(s)\,ds \leq (b- a)\sup_{s \in [a,b]} F(s). By extension, given I = \int_0^1 \sum_{i=1}^n\|\mathbf{F}_i(t,s)\||x_i - y_i|\,ds we can obtain \begin{split} I &\leq \sum_{i=1}^n \sup_s \|\mathbf{F}_i\||x_i -...
11. ### I Can you transform a countably infinite set to an uncountable one?

"All roots of natural numbers" might also include, for example, i, -1 and -i as fourth roots of 1, so its not that simple. Even if we restrict attention to real roots, we should also include -1.
12. ### Separable first order ODE involving tangent

I think from (\log |\sin x|)' = t the next step is \log |\sin x| = \log |A| + \tfrac12 t^2 and hence |\sin x| = |A|e^{\frac12 t^2}. It follows from this that \sin x and A have the same sign, so we can drop the absolute value signs.
13. ### Find the limit of the sequence

It's not sufficient to note the existence of a fixed point of the iteration. The fixed point might be unstable, or if it is stable then the initial value might not be in its domain of stability. Those things need to be checked. In this case, you can show that for any x \geq 0, \left|\sqrt{2x}...
14. ### I Result of this integral in large Lambda limit

To continue my earlier post, from \cosh v_0 = \frac{1 + \epsilon^2}{1 - \epsilon^2} we can obtain \begin{split} \epsilon^2 &= \frac{\cosh v_0 - 1}{\cosh v_0 + 1} \\ &= \frac{2\sinh^2(v_0/2)}{2\cosh^2(v_0/2)} \\ &= \tanh^2(v_0/2)\end{split} so that \frac{2\epsilon^2}{(1 - \epsilon^2)^2}\ln...
15. ### I Result of this integral in large Lambda limit

We have \int_0^\infty \frac{u}{(u + \epsilon^2)(1 + u)^2}\,du = \int_0^\infty \frac{\epsilon^2}{(1 - \epsilon)^2}\left(\frac{1}{1 + u} - \frac{1}{u + \epsilon^2}\right) + \frac{1}{1 - \epsilon^2}\frac{1}{(1 + u)^2}\,du. There is in fact no difficulty with \begin{split} \int_0^\infty \frac{1}{u...
16. ### I Result of this integral in large Lambda limit

Susbstituting q = \Lambda x = \Lambda \sqrt{u} gives \begin{split} \int_0^\infty \frac{q^3}{(q^2 + m^2)\left(1 + \frac{q^2}{\Lambda^2}\right)^2}\,dq &= \Lambda^2 \int_0^\infty \frac{x^3}{(x^2 + \epsilon^2)(1 + x^2)^2}\,dx \\ &= \frac{\Lambda^2}{2} \int_0^\infty \frac{u}{(u + \epsilon^2)(1 +...
17. ### B Would like some more knowledge about the product of functions

You know that (2 + \sin (x/2))^2 > 0 for any x. So y can only be negative when \cos(x/2) < 0, which is not the case for x \in [0, \pi].
18. ### A question about definite integrals and series limits

\left| a_n - \frac{\pi}{12}\right| = \int_0^{2-\sqrt{3}} \frac{x^{4n}}{1 + x^2}\,dx < \int_0^{2-\sqrt{3}} x^{4n}\,dx = \frac{(2 - \sqrt{3})^{4n+1}}{4n+1} is a tighter bound; depending on what you're doing convergence as (4n+1)^{-1} may not be adequate.
19. ### Solve the given simultaneous equations

If x + y = 2a and xy = a^2 then x and y are the roots of \begin{split} 0 &= (z - x)(z - y) \\ &= z^2 - (x + y)z + xy \\ &= z^2 - 2az + a^2 \\ &= (z - a)^2.\end{split} Thus (x,y) = (a,a) is the only possibility.
20. ### A question about definite integrals and series limits

Are you sure about that? What you have written for the numerator is 1 - x^{4n} = (1 - x^{2n})(1 + x^{2n}). Did you mean (1 - x^4)^n? If so, you end up with a polynomial in x, which you can integrate analytically: \begin{split} a_n &= \int_0^{2 - \sqrt{3}} (1 - x^4)^{n-1}(1 - x^2)\,dx \\ &=...
21. ### Evaluate Indefinite Integrals

You can't integrate \int (t - 2)^2\sqrt{t}\,dt = \int t^{5/2} - 4t^{3/2} + 4t^{1/2}\,dt?
22. ### I Why didn't Leavitt sign the paper where the discovery of Leavitt's Law is reflected?

Sadly, I suspect the answer is that it is because she was female.
23. ### Evaluate Indefinite Integrals

Did you consider t = x + 2, dx = dt?
24. ### I Finding the derivative of a characteristic function

You can almost always rewrite \sum_{k=-\infty}^{\infty} a_k = a_0 + \sum_{k=1}^\infty (a_k + a_{-k}). To calculate the derivative, I would abstract the details until they are needed. So start with F(t) = \left(\sum_{k=-\infty}^{\infty} f_k(t)\right)^n and proceed to F'(t) =...
25. ### I Solving Differential Equation Using Reduction of Order

I don't agree with your result for v. Your equation for v' can be reduced to \begin{split} 0 &= xy_1v'' + (2xy_1' - y_1)v' \\ &= \frac{x^2}{y_1} \left( \frac{y_1^2}x v'' + \left( \frac{2y_1y_1'}{x} - \frac{y_1^2}{x^2}\right)v'\right) \\ &= \frac{x^2}{y_1} \frac{d}{dx}\left( \frac{y_1^2...
26. ### Symmetry of an Integral of a Dot product

Easier is \begin{split} \int_0^{2\pi}\int_0^{2\pi} \cos(\phi - \phi')\,d\phi\,d\phi' &= \int_0^{2\pi}\left[ \sin(\phi-\phi') \right]_{\phi=0}^{\phi = 2\pi}\,d\phi' \\ &= \int_0^{2\pi}\sin(2\pi - \phi') - \sin (-\phi')\,d\phi' \\ &= \int_0^{2\pi}0\,d\phi' \\ &= 0.\end{split}
27. ### Symmetry of an Integral of a Dot product

In terms of plane polar coordinates (r,\phi) and (r,\phi') you have \theta = \phi - \phi'. To integrate over a circle in the (r',\phi') you must not only integrate with respect to r' from 0 to R, but also with respect to \phi' between 0 and 2\pi. The integral of \cos(\phi - \phi') over a...
28. ### POTW Estimate of a Principal Value Integral

I assume by A(x) you mean \operatorname{Ai}(x) or vice-versa.

30. ### A Stability analysis for numerical schemes of systems of PDEs

In principle, you can use the same technique: define the discrete fourier transforms of both variables, \begin{split} \hat\nu(\zeta, t) &= \sum_{n=-\infty}^\infty \nu_n(t)e^{in\zeta} \\ \hat u(\zeta, t) &= \sum_{n=-\infty}^\infty u_n(t)e^{in\zeta}\end{split} Then taking the DFT of your PDEs you...
31. ### Find the derivative of the given function

I'd just like to note that, in the proposed solution by taking logs before differentiating, one should first simplify \ln(x^5) = 5 \ln x and \ln(\sqrt{x^2 + 2}) = \frac12\ln(x^2 + 2) before taking the derivative, thereby saving an application of the chain rule.
32. ### On the ratio test for power series

It is fairly easy to show from the definition of the limit of a sequence and the definition of a sequence diverging to +\infty that \lim_{n \to \infty} |b_n| = \infty\quad\Rightarrow\quad\lim_{n \to \infty} \frac{1}{|b_n|} = 0 and \lim_{n \to \infty} |b_n| = 0 \quad\Rightarrow\quad \lim_{n \to...
33. ### MHB Solving Quadratic Equations without CD: Better Direction?

Set u = x + x^{-1}. Then x^2 + x^{-2} = u^2 - 2 and x^3 + x^{-3} = u^3 - 3u so that u^3 + u^2 - 2u - 30 = (u- 3)(u^2 + 4u - 10) = 0. Then completing the square in x gives (2x - u)^2 = u^2 - 4 and the choice u = 3 leads directly to (2x - 3)^2 = 5. The other roots u = -2 \pm \sqrt{14} lead to...
34. ### I Taylor series of 1/ln(t+1) at t=0

1/\log(1 + t) doesn't have a Taylor series about t = 0, because the function is not defined (and therefore not differentiable) there: \log(1 + 0) = \log 1 = 0. It does have a Laurent series about t = 0.
35. ### POTW Positive Definiteness Determined from Symmetrized Products

Note that as P and Q are self-adoint on a finite-dimensional complex inner product space (V, \langle \cdot, \cdot \rangle), their eigenvalues are real and they each have a basis of orthogonal eigenvectors. By definition, a linear map L: V \to V is positive definite if and only if \langle Lx ,x...
36. ### Does a linearly time varying B create changing E?

You have \mathbf{B} = \mathbf{b}(\mathbf{x})t where \nabla \cdot \mathbf{b} = 0. Hence in the absence of sources, (\nabla \times \mathbf{b})t = \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} so that \mathbf{E} = \frac{c^2t^2}{2} (\nabla \times \mathbf{b}) + \mathbf{E}_0(\mathbf{x}). From...
37. ### I Use the Calculus of Variations to show the minimum path connecting two points is a straight line

In this case, \frac{\partial f}{\partial y} means differentiation of f with respect to its first argument, and \frac{\partial f}{\partial y'} means differentiation of f with respect to its second argument, in both cases with the other argument held constant. You are differentiating not f, but...
38. ### Show function series involving arctan is not differentiable at x=0

If f_K(x) = \sum_{k=1}^K \frac{\arctan(kx)}{k^2} then the series expansion f_K(x) = \sum_{k=1}^K \sum_{n=0}^\infty \frac{(-1)^nk^{2n-1}x^{2n+1}}{2n+1} is only valid for x\in \bigcap_{k=1}^K \{t \in \mathbb{C} : k|t| < 1\} = \{ t \in \mathbb{C}: |t| < K^{-1}\} so that the radius of convergence...
39. ### Show function series involving arctan is not differentiable at x=0

Another approach is proof by contradiction: Assume that f'(0) exists and see what follows.
40. ### Show function series involving arctan is not differentiable at x=0

We know that f(0) = 0, so it suffices to show that \lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to 0} \lim_{K \to \infty} \sum_{k = 1}^{K} \frac{\arctan(kx)}{kx} \frac 1k does not exist. That \lim_{y \to 0} \frac{y}{\tan(y)} = 1 might be useful.
41. ### How many positive integers x satisfy this logarithmic inequality?

Assuming your logs are to base 2 then this is one of the possible solutions. It is convenient to use logs to base 2; then the left hand side is \log_{x/8}(x^2/4) = 2\frac{ \log_2(x/4)}{\log_2(x/8)} = \frac{2 (\log_2 x - 1)}{\log_2 x - 3} and the right hand side is 7 + \log_2(8/x) = 10 -...
42. ### Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

Let f_i(x) = (1 - x)^i - (1 -2x)^i with x \in [0,1]. This is continuous with f_i(0) = 0 for all i \geq 1, so the problem comes for large k.

See post #3.
44. ### Show that the Taylor series for this Lagrangian is the following...

I'm not sure that's correct, or at least the notation is confusing. This is incorrect. \epsilon here is playing the role of x and v is playing the role of a, but the role of f is played by (v \mapsto L(v^2)) rather than L. So we should find \begin{split} L(v^2 + 2\epsilon v + \epsilon^2) &=...
45. ### I Order of an element in ##\mathbb{Z}_n##

Quicler is, for positive integers n and m, 12n - 20m = 4(3n - 5m) = 0\quad\Leftrightarrow\quad (n,m) = (5k,3k), k \in \mathbb{N} and the order of 12 in \mathbb{Z}_{20} is found by taking k = 1.
46. ### A Calculate a tensor as the sum of gradients and compute a surface integral

Your error is in stating that \vec \omega \times \vec r = r \omega \sin \theta. That deals with the magnitude of the cross-product, but not its direction; that must also depend on \phi. EDIT: Even the magnitude is only correct when \sin \theta \geq 0; outside of this range you must multiply by...
47. ### A Calculate a tensor as the sum of gradients and compute a surface integral

Your integral is f(r)r^2 \int_0^{\pi} \sin \theta \int_0^{2\pi} (\vec \omega \times \vec r)\,d\phi\,d\theta. Without loss of generality, you can assume \hat\omega = \omega \hat z. Then \vec \omega \times \vec r = \omega(x \hat y - y \hat x). Integrating either x = r \cos \phi \sin \theta or y =...
48. ### I Heat Equation: Solve with Non-Homogeneous Boundary Conditions

What do we know about T_{\mathrm{air}}? Have you tried a Laplace transform in time? That gives \kappa(p\hat T - T_0) = \frac{\partial^2 \hat T}{\partial x^2} where \kappa= \rho c_p subject to \hat T(0,p) = \frac{T_s}{p}, \quad \left.-k\frac{\partial \hat T}{\partial x}\right|_{x=L} = h(\hat...
49. ### A Solving System of PDEs

The OP did state "While trying to solve problem in Hydrodynamic stability ...". However, since the OP hasn't told us which problem they are trying to solve or how they derived this system, we can but speculate. I did misinterpret the situation in my earlier post. The system of PDEs looks like...
50. ### A Solving System of PDEs

This is a linearised stability problem, so you are looking for normal modes. Your assumption, in view of the boundary conditions, is that (\theta, \psi) = (\Theta e^{k_nx}\sin( n\pi y), \Psi e^{k_nx}\sin (n \pi y)) for constant (\Theta, \Psi) and positive integer n. If k_n has strictly...