Inverse Definition and 1000 Threads

Homework Statement Doing some exam revision and one of the questions from an old exam has me stuck at the last step, simply need to inverse the following F( \omega ) = \frac{e^{i \omega}}{1+\omega ^2} We're allowed to use a table on the exams but I cannot find anything quite...

Say we want to differentiate \arcsin x. To do this we put y=\arcsin x. Then x=\sin y \implies \frac{dx}{dy}= \cos y. Then we use the relation \sin^2 y + \cos^2 y = 1 \implies \cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - x^2}. Therefore \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}. My question is that...

I can derivate x(y) wrt y using the derivative of y(x) wrt x, follows the formula: \frac{dx}{dy}=\frac{1}{\frac{dy}{dx}} until same the 2nd derivative (taking the 2nd diff form of x and deriving wrt to x):d^2x=\frac{d^2 x}{dy^2} dy^2 + \frac{dx}{dy} d^2y \frac{d^2x}{dx^2}=\frac{d^2 x}{dy^2}...

So, I'm doing a problem where I take arctanh to a limit, and I was wondering if the arctanh function goes to infinity if the argument inside of the function goes to infinity when passing through the limit.

Prove that if operator on a hilbert space $T$ commutes with an operator $S$ and $T$ is invertible, then $T^{-1}$ commutes with $S$. $T^{-1}S$=$T^{-1}T^{-1}TS$=$T^{-1}T^{-1}ST$

Let $T_{n}$ be a sequence of invertible bounded linear operators with limit $T$ Prove that $(T_{n})^{-1}$ tends to $T^{-1}$

Homework Statement d/dθ csc-1(1/2)^θ = ? Homework Equations d/dx csc-1(x) The Attempt at a Solution I don't know how to deal with the exponent θ

When I place the trigonometric functions in the "wolfram google", it informs the parity of the function, so, sin(x), sinh(x) -> odd cos(x), cosh(x) -> even tan(x), tanh(x) -> odd cot(x), coth(x) -> odd sec(x), sech(x) -> even csc(x), csch(x) -> odd arcsin(x), arcsinh(x) -> odd...

Homework Statement Prove/Disprove following function being one-to-one.If yes,find its inverse. g(x)=x-\frac{1}{x},x>0 Homework Equations The Attempt at a Solution My tutor said that it is one-to-one,but I found that the are two solutions for g-1(x). Are there any mistakes...

Hi, I would like to find the inverse Laplace transform for 11/(s^2+16)^2 I have tried to expand it using the following partial fraction decomp to find the constants and take the inverse Laplace but this did not work C1(s)+ C2/(s^2+16) + C3(s)+C4/(s^2+16)^2 Does anyone have any suggestions?

Homework Statement Find: Inverse Laplace for x(t)= (e)^-5t*(t)^4 using laplace table and laplace properties. Homework Equations The Attempt at a Solution Well, I have been working on this problem for a few days now and cannot seem to figure it out. The two functions are not...

Homework Statement For a group G consider the map i:G\rightarrow G , i(g)=g^{-1} For a subgroup H\subset G show that i(gH)=Hg^{-1} and i(Hg)=g^{-1}H Homework Equations The Attempt at a Solution I know that for g_1,g_2 \in G we have i(g_1g_2)=(g_1g_2)^{-1}=g_2^{-1}g_1^{-1} Then...

Homework Statement If * is a binary operation on a set B, and the domain of definition is B^2, if * is associative and the neutral element is p. If r and l are elements of b we can say that r is a left inverse of l under * iff r * l = p, and l is a right inverse of r iff l * r = p. Show that if...

Digital signal processing -- Pseudo Inverse Method Homework Statement The Attempt at a Solution (a) A =the matrix with [ .4 0 0 0 0 0 0 0 0 0 0 0; .7 .4 0 0 0 0 0 0 0 0 0; -.1 .7 .4 0 0 0 0 0 0 0 0;... all the way down to 0 0 0 0 0 0 0 0 0 0 -.1] so it is 11 x9 . w=[w0 w1 ... w8]'...

Find the mean square error using the pseudo inverse approach. I am given a 11X9 matrix A, a 11X1 vector F and R = 11X11 diagonal matrix so Rhat = A[(A'A)^-1 ]A' R . Then I get a 11X11 matrix. Shouldn't I get getting a 8X11 matrix How do I get the most optimum vector F?

This is for Calculus II. I've found most of the integrations on inverse trig functions to be pretty simple, but for some reason this one is throwing me off. Homework Statement \int\frac{x+5}{\sqrt{9-(x-3)^2}}dx The Attempt at a Solution I started by breaking the integral up...

To prove that sin^{-1}(ix)=2n\pi\pm i log(\sqrt{1+x^2}+x) I can prove sin^{-1}(ix)=2n\pi+ i log(\sqrt{1+x^2}+x) but facing problem to prove sin^{-1}(ix)=2n\pi- i log(\sqrt{1+x^2}+x) Help please

Homework Statement Is there a way to evaluate L^{-1}(\frac{F(s)}{s + a})? I'm sure if it can be evaluate. Homework Equations The Attempt at a Solution

I can't seem to part of an inverse Laplace transform correct. \begin{align*} f(t) &= \frac{6}{5}\mathcal{L}^{-1}\bigg\{\frac{1}{s + 2}\bigg\} + \frac{3}{5}\mathcal{L}^{-1}\bigg\{\frac{3s - 1}...

How do you find the inverse of metric tensor when there are off-diagonals? More specifivally, given the (Kerr) metric, $$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$ we have the metric tensor; $$ g_{\mu \nu} =...

Here is the question: I have posted a link there to this thread so the OP can view my work.

In the Math Challenge Forum it has been requested fo compute the series... $\displaystyle S = \sum_{n=1}^{\infty} \tan^{-1}\ \frac{\sqrt{3}}{n^{2} + n + 3}\ (1)$ ... and that has been performed using the general identity... $\displaystyle \sum_{n=1}^{\infty} \tan^{-1}\ \frac{c}{n^{2} + n +...

Just curious: Are there any unique relationships b/w the inverse of a function and the original, specifically when considering the derivative and integral?

If f and g are inverse functions and f ' is continuous, prove that: [from a to b] ∫ f(x) dx = b f(b) - a (a) - [from f(a) to f(b)] ∫ g(y) dy Hint: Use part (a) and make the substitution y = f(x) I have been trying to rearrange the equations and have looked online at answers, and still...

If g(a) \neq 0 and both f and g are continuous at a, then we know the quotient function f/g is continuous at a. Now, suppose we have a linear operator A(t) on a Hilbert space such that the function \phi(t) = \| A(t) \|, \phi: \mathbb R \to [0,\infty), is continuous at a. Do we then know that...

Hi, All: Let ## f: X → Y ## be a differentiable map , so that ## Df(x)≠0 ## for all ##x## in ##X##. Then the inverse function theorem guarantees that every point has a neighborhood where ##f ## restricts to a homeomorphism. Does anyone know the conditions under which conditions a map like...

Can someone explain why the answer is D a < 0 because it finishes downwards e < O because the y-intercept is in the negatives. b, & d = zero (but i don't get this) c is supposedly > 0 (nor do i get this) According to the solutions the graph is an even function, and symmetrical about the...

Hello, Homework Statement I would like to find the inverse Z transform of the following: F(z)=1-1.25z-1+0.25z-2/[1-(5/6)z-1+(1/6)z-2] using (a) partial fractions, and (b) residue theorem I have obtained different results and hence would appreciate some insight on the discrepancy and how...

I know potential has an inverse relationship with distance. However what is the equation that deduces this?

Homework Statement Find ##(g_ig_j)^{-1}## for any two elements of group ##G##. Homework Equations For matrices ##(AB)^{-1}=B^{-1}A^{-1}## The Attempt at a Solution I'm not sure how to show this? I could show that for matrices ##(AB)^{-1}=B^{-1}A^{-1}##. And that for numbers...

We have a matrix with dimension NxN.For some m belongs to N,m0 we have A^m0=0.We consider the exponential matrix e^A=I+A+A^2/(2!)+A^2/(3!)+A^m/(m!).Find the inverse matrix of e^A. I tried to write the e^A=e^A(m0)+A^m/(m!) or (e^A)^(-1)=(...

I'm looking for solutions to this problem: Matrices A(m,n) and B(n,m) satisfy AB=I(m,m) where n isn't equal to m. Can I find a matrix S(m,n) such that SA=I(n,n) or SA approximates I(n,n)? By approximate I don't have preferred definition, hence any suggestion is welcome!

Homework Statement ##\mathcal{L}^{-1}\Big\{\frac{s}{(s^2+1)^2}\Big\}## I'm trying to figure out how to find the inverse Laplace transform of this expression. Is this something you just look up in a table or is there a way to find it directly, maybe by Convolution?

Hi, a long time ago, back in high school, I remember my teacher was explaining the force of gravity to us. He gave us the equation for the force of gravity, which was proportional to the inverse square of the distance. However, he later said that something about Einstein and other researchers...

Homework Statement A patrol car is 50 ft from a long warehouse. The revolving light on top of the car turns at a rate of 30 rotations per minute. How fast is the beam of light moving along the warehouse wall when the beam makes a 45° angle with the line perpendicular from the light to the...

Is it just ∇-1 with the vector hat?

Suppose we are given two projection operators H' and H'' such that H' + H'' = 1, i.e. that any vector can be written as V = V' + V'' = (H' + H'') V. In a formula for a projection of the Riemann tensor (see the thread "Projection of the Riemann tensor formula") I encountered the term...

Homework Statement Given the function f(x) = (abs(x))*x +6, find f^-1(x) Homework Equations The Attempt at a Solution for x≥ 0, f(x) = x^2 + 6 y=x^2 +6 x = √(y-6) for y≥6 → f^-1(x) = √(x-6) for x≥6 for x< 0, f(x) = -x^2 + 6 y= -x^2 +6 x = √(6-y) for y<6 → f^-1(x) = √(6-x) for x<6 But...

For the given function i have to find if is surjective/injective and find the inverse of the function: f(x)=\frac{3x-2}{x+2} I now that for inverse i have to express $x$ somehow,but i don't know how to do it... Thank you for the help!

Inverse trig problem -- please help! Homework Statement tanx+tan2x+root3tanxtan2x=root3 find x... Homework Equations The Attempt at a Solution i have tried a lot and always ended with a complicated cubic equation... please help me by giving me a another approach to the solution

Here is the question: I have posted a link there to this thread so the OP can see my work.

Hello, I have another question regarding inverse matrices. I need to find \[a^{2}\] given: \[\exists x: \begin{pmatrix} 1 &a \\ 2a &1 \end{pmatrix}^{2}\cdot \begin{pmatrix} 1\\ x \end{pmatrix}=\begin{pmatrix} 0\\ 0 \end{pmatrix}\] Any hints or guidance will be appreciated ! Thanks !

Hello all, I have this matrix A \[A=\begin{pmatrix} 1 &2 &3 &4 \\ 9 &8 &2 &0 \\ 17 &2 &0 &0 \\ 1 &0 &0 &0 \end{pmatrix}\] B is defined as the inverse of A. I need to find the element in the first row and fourth column of B, without using determinants, so without using adjoint. How should I...

prove the function ## g: \mathbb{N} \rightarrow \mathbb{N} ## ## g(x) = \left[\dfrac{3x+1}{3} \right] ## where ## [y] ## is the maximum integer part of r belonging to integers s.t. r less than or equal to y is surjective and find it's inverse I know this function is bijective, but how do I...

Hello MHB, I am aware of there is two way, u can use chain rule or defination of derivate. I totaly understand the proof with this type Derivative of Inverse Function but is that a valid proof? How ever our teacher did proof this with derivate defination which I don't understand from my...

I confront an integration with the following form: \int d^2{\vec q} \exp(-a \vec{q}^{2}) \frac{\vec{k}^{2}-\vec{k}\cdot \vec{q}}{((\vec q-\vec k)^{2})(\vec{q}^{2}+b)} where a and b are some constants, \vec{q} = (q_1, q_2) and \vec{k} = (k_1, k_2) are two-components vectors. In the...

Find the Inverse Laplace of 1/(s^3) is there some special rule for cube? The answer is t^2/2 Looking at the Laplace Table t^n looks similar but its not it exactly. What should I do?

Assume that f: E \to Y \,\,\, , E \subset X then can we say that f(E^c)=f(E)^c what about the inverse mapping f^{-1}: V \to X \,\,\, , V\subset Y do we have to have some restrictions on f and its inverse ? My immediate answer is that we have to have a bijection in order to conclude that but I...

Homework Statement If f(x) = the third root of (x-8), find the derivative of its inverse. Homework Equations The derivative of its inverse = 1/f'(f^-1(x)) or 1 over its derivative at its inverse. The Attempt at a Solution I followed both the formula to verify my solution and...

Homework Statement I wish to prove that for s>1 $$ \sum\limits_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}. $$ The Attempt at a Solution (1) I first showed that $$ \prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}. $$ It was a given theorem in the text that $$...