Inverse Definition and 1000 Threads

Yesterday, I was thinking about a problem I had encountered many years before, the central force problem with a ##V(r) \propto r^{-2}## potential... If we have a Hamiltonian operator ##H = -\frac{\hbar^2}{2m}\nabla^2 - \frac{A}{r^2}## and do a coordinate transformation ##\mathbf{r}...

What is the formula to solving a problem like this? Thanks in advance!

Homework Statement There is a signal y[n] with a differentiable DTFT Y(eiw). Find the inverse DTFT of i(d/dw)Y(eiw) in terms of y[n] (where of course i = √-1). Homework Equations Sifting property ∫eiwndw = 2π*δ[n] from [-π,π] (integral a) leads to ∫Y(eiwn)dw = 2π*y[n] from [-π,π] (integral b)...

Given a unit-hypotenuse triangle, how do we get the inverse sin/cos/tan equations? I'm trying to program a high-precision fixed-fraction model of the sun and Earth and I've forgotten how the equations are derived. I know there's differentiation and integration. And I'm stuck on how to express...

Hello, I have tried the integral below with Mathematica and it gives me the following solution: ##\frac{d}{dc}\int_{z^{-1}(c)}^{1} z(x)dx = -\frac{c}{z'(z^{-1}(c))}## I am not quite sure where it gets it from...I think it can be separated and with differentiation the first part will be zero...

I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...

This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...

Homework Statement Suppose ##f(x) = x^5 + 2x + 1## and ##f^{-1}## is the inverse of function f. Evaluate ##f^{-1}(4)## solution: 1/7 Homework Equations ##(f^{-1}(x))=\frac{1}{f'(f^{-1}(x))}## The Attempt at a Solution I attempted to use my calculator's solve function to get the solution of...

Homework Statement ƒ(s) = 1/((1-exp(-s))*(1+s)) Homework EquationsThe Attempt at a Solution I know the solution is periodic but how to obtain the t-domain function?

$\tiny{242.7x.01}$ $\textsf{Find the value of $df^{-1}/dx$ at $f(a)$, $x\ge4$, $a=3$}$ \begin{align*}\displaystyle f(x)&=x^3-6x^2-3 \\ \frac{df^{-1}}{dx}\biggr\rvert_{3} &=\frac{1}{{\frac{d}{dx}}\biggr\rvert_{3}} \\ &=\frac{1}{3x^2-12x} =\frac{1}{3(3)^2 - 12(3)}\\...

Homework Statement [/B] Given this matrix ##\begin{bmatrix}As+B \\ C \end{bmatrix}## which is invertible and ##A## has full row rank. I would like to show that its inverse has no terms with ##s## or higher degree if ##\begin{bmatrix}A \\ C \end{bmatrix}## is invertible. Homework Equations...

Homework Statement Determine the inverse Laplace transform Homework Equations 3s+9. (s+3)^2+7 The Attempt at a Solution [/B] Hi iam new to the forum and still unsure how to make the equations the correct format so hope you can understand what I have typed. I have Tried to Convert...

Homework Statement Determine the inverse Laplace transform Homework Equations 3s+9/(s+3)^2+7 The Attempt at a Solution Converted to 3s+9/s^2+6s+16 to try and use the partial fractions method but getting nowhere. I'm Not sure if Iam making the question more difficult, can't seem to put the...

Homework Statement I want to invert a function from Laplace transform space to normal space. Homework Equations In Laplace transform space, the function takes the form $$ \bar f (s) = \frac{\exp\left[ x (-a +\sqrt{a^2+ b +c s} )\right]}{-a +\sqrt{a^2+ b +c s}}. $$ Here, ##s## is the Laplace...

Hello I have a question about the uniqueness of the inverse element in a groupoid. When we were in class our profesor wrote ##\text{Let} (M,*) \,\text{be a monoid then the inverse (if it exists) is unique}##. He then went off to prove that and I understood it, however I got curious and started...

Suppose we have a product formed by a multiplication of a unitary matrix U and a diagonal matrix A, can we retrieve the inverse of A without knowing either U or A?

Homework Statement Homework Equations N/A The Attempt at a Solution The left hand side (red box) is the data sheet provided to us in the exam. The right hand side (blue box) is Wolfram Alpha. The data sheet says that the inverse Laplace transform of 1/s is equal to u(t) (i.e. the unit step)...

I went to Amadeus Live yesterday at the Eugene (Oregon) Symphony place (Hult Center). It was an interesting combination of the film, Amadeus (1984) with live music played by the symphony with a chorus (a trans-time connection). They pulled it off pretty well. Sometimes the music was a bit too...

Homework Statement Solve ut+3ux=0, where -infinity < x < infinity, t>0, and u(x,0)=f(x).Homework Equations Fourier Transform where (U=fourier transform of u) Convolution Theorem The Attempt at a Solution I've used Fourier transform to get that Ut-3iwU=0 and that U=F(w)e3iwt. However, I'm...

Homework Statement The two graphs are possible legitimate representations of ##y=\sec^{-1}(x)##. The derivative is positive on all the domain and so is graph A, but graph B has negative tangent when x<-1 Homework Equations Derivative of inverse secant...

Homework Statement T/F: Each eigenvector of an invertible matrix A is also an eignevector of A-1 Homework EquationsThe Attempt at a Solution I know that if A is invertible and ##A\vec{v} = \lambda \vec{v}##, then ##A^{-1} \vec{v} = \frac{1}{\lambda} \vec{v}##, which seems to imply that A and...

Homework Statement $$\int\frac{dx}{x\sqrt{x^2-1}}=\left\{ \begin{array} {lc} \sec^{-1}(x)+C_1 & {\rm if}~x>1 \\ \sec^{-1}(-x)+C_2 & {\rm if}~x<-1 \end{array} \right.$$ Why the second condition ##\sec^{-1}(-x)+C_2~~{\rm if}~x<-1## ? Homework Equations Derivative of inverse secant...

Hello everybody, If I define z_\mu = \frac{\partial{\phi}}{\partial{x^{\mu}}}, \, \mu = 0,1,...,n , (for some scalar function phi of x=(x_0,...,x_n)) how is then \frac{\partial{}}{\partial{z_{\mu}}} defined or rather what is it equal to? How would you call this expression? the inverse of a...

Homework Statement F(s) = \frac{s^2e^{-4s}}{s^2+9} Homework Equations \mathcal{L} ({f(t-a)\mathcal{H}(t-a)}) = e^{-as}F(s) The Attempt at a Solution F(s) = e^{-4s}(\frac{s^2}{s^2+9}) using \mathcal{L} {f(t-a)\mathcal{H}(t-a)} = e^{-as}F(s) f(t) = cos(3t-12)\mathcal{H}(t-4) I do...

Hi, everyone, the question is as below: Find the inverse Laplace transform to 1/(350+s) * X(s). 's' is the Laplace variable and 'X(s)' is also a variable. I inverted 1/(350+s) and X(s) separately and multiplied them together directly. But this seems not giving me the correct answer. Could...

Homework Statement Snell's law is: $$\frac{\sin\theta_1}{c_1}=\frac{\sin\theta_2}{c_2}$$ $$\frac{c_1}{c_2}=n_{12}$$ Express ##\theta_2## as a function of ##\theta_1## Find the largest value of ##\theta_1## for which the expression for ##\theta_2## that you just found is...

How do I go about find out the inverse of this giant $7 \times 7$ matrix? Do I need to evaluate it using the tedious Gauss-Jordan method? Any property, short-cut, trick that I can take advantage of? $$\begin{pmatrix} 0 &1 &0 &0 &0 &0 &0\\ 0 &0 &0 &1 &0 &0 &0\\ 0 &0 &0 &0 &0 &0 &1\\ 0 &0 &0 &0...

Homework Statement Homework Equations determinant is the product of the eigenvalues... so -1.1*2.3 = -2.53 det(a−1) = 1 / det(A), = (1/-2.53) =-.3952 The Attempt at a Solution If it's asking for a quality of its inverse, it must be invertible. I did what I showed above, but my answer was...

Homework Statement Simplify: $$\sin^{-1}(2\sin^{-1}0.8)$$ Homework Equations Inverse sine: ##y=\sin^{-1}(x)~\rightarrow~\sin(y)=x## $$\sin^2(x)+\cos^2(x)=1$$ The Attempt at a Solution The inner parenthesis: ##\sin y=0.8## . In the drawing it's alpha's sine. Now i double the α and the...

Homework Statement Let t = f(g, h, A, Δm, Γ, r). For t = 2 s, the propagated error is σ = 0.02 s. Can the error of 1/t2 be simply determined using the known error in t = (2 ± 0.02) s, or must the variance formula (with all the partial derivatives and errors of each dependent variable) be...

I need to prove that: $ \arctan{\dfrac{1}{x}}=\dfrac{\pi}{2}- \arctan{x}, \forall x>0$. Now, I assumed $\arctan{\dfrac{1}{x}}=\arccot{x}$. So, I've tried to do this: $\cot{y}=x \implies y=arccot{x} \\ \tan{y}=\dfrac{1}{\cot{y}}=\dfrac{1}{x} \implies y=\arctan{\dfrac{1}{x}} \\ \implies...

Homework Statement Consider the following matrix. A = 2 + 4i...1 + 5i 2 − 3i...2 + 3i Let B = A-1. Find b12 (i.e., find the entry in row 1, column 2 of A−1) Homework Equations A-1 = 1/(ad - cb)* [ d -b ] [ -c a ] <--imagine as 2x2 matrix with first row (d,-b) and second row...

Homework Statement A magnetic data set is believed to be dominated by a strong periodic tidal signal of known tidal period\Omega The field strength F(t) is assumed to follow the relation: F=a+b\cos\Omega t + c\sin\Omega t If the data were evenly spaced in time, then Fourier analysis would...

Homework Statement Suppose that a square matrix ##A## satisfies ##(A - I)^2 = 0##. Find an explicit formula for ##A^{-1}## in terms of ##A## Homework EquationsThe Attempt at a Solution From manipulation we find that ##A^2 - 2A + I = 0## and then ##A(2I - A) = I##. This shows that if we...

Temperatures can be converted from Fahrenheit to Celsius using the function f(x) = 5 /9 (x − 32). (a) Calculate f(59). (b) Find f −1 (x), and verify that f −1 (f(59)) = 59. (c) Let K be the set {x : f(x) = x}. Find all elements of K and list K

I have the following laplace function F(s) = (A/(s + C)) * (1/s - exp(-sα)/s)/(1 - exp(-sT)) I think that the inverse laplace will be- f(t) = ((A/C)*u(t) - (A/C)*exp(-Ct)*u(t)) - ((A/C)*u(t-α) - (A/C)*exp(-C(t-α))*u(t-α)) and f(t+T)=f(t) Now I want to find the Fourier series expansion of f(t)...

Homework Statement For school, I have to make a task about sound intensity and the distance to the sound source. I have to prove that the relation between these two is known as the inverse square law _1/ I_2 = ( _2/_1 )². Does someone know how I can plot the inverse square law or prove that it...

Homework Statement I have the second order diff eq: Solving by Laplace transform gets me to: I could use the inverse laplace transform that takes me back to e^{at}cos(bt) with b=0, but that only solves for the homogeneous (complementary) part of the equation, it won't reproduce the dirac...

Here is the question: This is the step I came to after taking the derivatives and doing some simplification: ^ I did the work myself on paper, I just couldn't type out the whole thing clearly so that anyone else can see what I'm referring too... so I used some online tool to show that...

I am doing independent research to compute eye strain and I hit a problem. Hope someone can help. This is the situation. Work done to pull a spring is ½kx2. The human lens is like a spring. Linear, and obeys Hooke’s law. Work done to stretch the lens is also ½ kx2 (k=spring constant of...

Please refer to the below image (Example 5). Do anyone know how 5x^4 > 1 > 0?

Homework Statement find the inverse Laplace transform of the given function by using the convolution theorem Homework Equations F(s) = s/((s+1)(s2)+4) The theorem : Lap{(f*g)(t)} = F(s)*G(s) The Attempt at a Solution I know how to find it the answer is : we have 1/(s+1) * s/(s+4) and the...

When, in wireless communications, does the inverse fourth power-law become relevant? My understanding is that is that what cause the average signal power to degrade to the forth power is cancellation from self reflections. So by my way of thinking, an LOS point to point system, like a...

Homework Statement *Main ideas in bold[/B] Investigation of the inverse square law of light radiated from a light bulb. (done method, diagram, results and graph) Independent variable = the distance from the LDR (cm) Dependent Variable = resistance (k/ohms) Brief method: using an LDR, bulb...

Homework Statement a.)y=sqrt(2^x -1) . I tried: b.)y=log(sqrt(2^x -2)) and c.)y=log^3 (2-sqrt(x)). Homework EquationsThe Attempt at a Solution x=sqrt(2^y -1) x^2 = 2^y -1 2^y = x^2 +1 y=log2(x^2 +1) y=2 log2(x+1) is that correct result? regarding b and c I am just lost :/. Will appreciate...

I'm looking for the inverse functions of the Maxwell-Boltzmann distribution and Planck's Law. Planck's Law in terms of the wavelength. Any of you know of any literature on this topic?

I am trying to solve with Laplace Transforms in an attempt to prove duhamels principle but can't find the Laplace transform inverse at the end. The book I am reading just says "from tables"... The problem : $$ U_t = U_{xx}\\\\ U(0,t)=0 \quad 0<t< \infty\\\\ U(1,t)=1\\\\ U(x,0)=0 \quad...

Homework Statement Let ##S ## be a cylinder defined by ##x^2 + y^2 = 1##, and given a parametrization ##f(x,y) = \left( \frac{x}{ \sqrt{x^2 + y^2}}, \frac{y}{ \sqrt{x^2 + y^2} },\ln \left(x^2+y^2\right) \right)## , where ##f: U \subset \mathbb R^2 \rightarrow \mathbb R^3 ## and ## U = \mathbb...

Homework Statement Hello All via the following x, y values , a formula needs to be decided for the 'Y ' variable. Basicaly Consider the table below for x and y (are they directly proportional or are they inversely related ? ). and also to find out the corresponding formula. Pl. see the...

{a,-a,0} ∈ Z For a set to have inverse under an operation, all elements must be able to be combined with another element of the set under that operation, after which the product of combination under said operation yields the identity element of that operation. My question is, this is a...