## Proof of Higher Order Inverse Differential Operator March 23

I'm having difficulty proving the following :$\frac{1}{D^2+\alpha^2}sin(\alpha x) = \frac{-x}{2\alpha}cos(\alpha x)$I know it won't use the transformation $P(D^2:\mapsto -\alpha^2)$, since it will put a zero in the denominator, and that it should inv

## How to solve the Matrix Equation with parameter $\lambda \in \mathbb{R}$ March 23

For given a matrix $A \in \mathbb{R}^{n \times n}$ and $\lambda \in \mathbb{R}$, Considering the following Matrix Equation (A^{T}A + \frac{\lambda I_{n}}{2}) X + A^{T} X A^{T}+ A X A + X (A^{T}A + \frac{\lambda I_{n}}{2}) = 0, \end{e

## on the necessity of gluing conditions March 23

Suppose we are given a family of schemes $\left\{X_i\right\}_i$, with $U_{ij}$ open in $X_i$ such that there exists isomoprhism $\phi_{ij}: U_{ij} \rightarrow U_{ji}$. Why do we need the condition $\phi_{ij}(U_{ij} \cap U_{ik}) = U_{ji} \cap U_{jk}$

## What exactly are the numbers we use everyday March 23

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its circumference". You need to define what a circle is. Wikipedia defines a circle as: It is

## Proving an integration with a modified Bessel function and an exponential March 23    1

I am trying to prove the following identity:where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of ...

## Proving the transform of the Q-function March 23    1

I have the Gaussian Q-function, given by:and I want to prove that it can be also expressed as:Can somebody help explaining how to obtain the second integral from the first?Both expressions tend to zero as $x\to +\infty$, so it is sufficient to show t

What is the Fourier trasnform of the function$$\frac{\sin(P|\mathbf{x-y}|)}{|\mathbf{x-y}|}$$where $P$ is a real parameter and $\mathbf{y}$ is a fixed point in three-dimensional space?It is a function with support on a sphere of radius $P$: $$\frac ## Deriving the addition formula for the lemniscate functions from a total differential equation March 23 2 The lemniscate of Bernoulli C is a plane curve defined as follows.Let a > 0 be a real number. Let F_1 = (a, 0) and F_2 = (-a, 0) be two points of \mathbb{R}^2. Let C = \{P \in \mathbb{R}^2; PF_1\cdot PF_2 = a^2\}. Then the equation of C ## Looking for a function that fits a certain criteria March 23 2 I am looking for a function that fits this description:$$ \frac{d^n}{dx^n}[f(x)] = n! f(x) $$or$$ \frac{d^n}{dx^n}[f(x)] = (n-1)! f(x) $$For all values of n, with this function i am looking to derive a new identity, would be greatly appreciated.T ## Verification of Fourier transformation of Io-sinh function March 23 I try to match, but it could not match I_o-\sinh Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ... ## Integrating factors — how in the world does one calculate those March 23 1 Is there an easier way of computing an integrating factor for differential equations? I need help understanding how to calculate those. I know the reason for them but just not familiar with how to compute exponential power functions. Help pleaseYou n ## Create 'smooth breakpoint function' by using integral March 23 1 Experts, I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a function with certain properties. In biology, there is ## mathieu function of non integer order asymptotics March 23 I have an asymptotic expression for the integer mathieu functions: se_\nu(q,z) and ce_\nu(q,z), where \nu is an integer. I would like to use these expression for the case \nu real. My question is, is there some way to represent a non-integer ## Mathieu function rescale problem March 23 1 The Mathieu functions are the solutions for the equation$$ y''+(a-2q\cos(2z))y=0 $$If we require the solution has the form$$ y(z) = e^{i r z}f(z) $$where f(z) is a periodic function with ... ## Looking for a function which satisfies interesting property: March 23 I am looking for a function(s) which satisfy the following property:$$f(x/t)*f(-y/r)=f((x-y)/(t-r))$$I am not sure if there is any function which satisfies this property.I thought that using an ansatz like f(x/t)=exp(g(x/t)) should help. But I can' ## If I'm perpendicular to an object that is 1 mile long and traveling perpendicular to me March 23 If I know an objects real size and I get its apparent size by putting a ruler 1 foot from my eye and measuring it. If the object is perpendicular to my eye is that enough information to determine it's distance from me? What would the calculation be? ## How to prove that a-b \ge \sqrt3{ab} March 23 Let a and b be distinct positive integers such that ab(a+b) is divisible by (a^2+ab+b^2).How to prove that |a-b| \ge \sqrt[3]{ab}? I have no idea how to do this. Can this be proved with simple way? ## Initial value problem for a system of ODEs with two parameters March 23 2 Consider the following initial value problem for the autonomous system of ODEs %%\label{eqn: } \begin{cases} %\vspace{3mm} x'(t)=y(t),\; t>0,\\ %\vspace{3mm} y'(t)=-\frac{ ... You Might Also Like • Find an orthonormal basis for the subspace of \mathbb{R}^4 that consists of vectors perpendicular to u = ... • I want to prove convexity of the following function:$$f(x) = log_x \left(1 + \frac{(x^a-1)(x^b - 1)}{x-1}\ri ...
• Given a category $\mathcal{C}$ we have a class $\mathrm{Obj}(\mathcal{C})$ of objects of $\mathcal{C}$. Depe ...
• What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curio ...
• I have two equations;1) $$0 = 7x+13$$and2) $$0 = 2x-13$$Why can I use the false position/secant method in t ...
• In learning how to rotate vertices about an a ...
• The question is to find the value of:$n\choose 1$$x(1-x)^{n-1} +2.n\choose2$$x^2(1-x)^{n-2}$ + 3$n\choose3 ... •$(\forall x)(x^2+4x+5\geqslant 0)$universe is$\Re$I went about it this way$x^2+4x \geqslant -5x(x+4) \g ...
• I've been asked to write down a statement using predicate calculus and it is confusing me a great deal.I've ...
• Today in lectures we were doing a brief review of some metric spaces stuff and I'm not quite sure about some ...