Function Definition and 1000 Threads

The integral of cothx is ln|sinhx|+C. Does this mean the integral of coth2x is ln|sinh2x|+C? If not, does anyone have a link to a page on how it is achieved - I'm trying to compile a list of all common hyperbolic function derivatives and integrals. However, I can't find anything to confirm if...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.14...

Consider the following periodic function: ## f(t) = \sin(ωt) + \cos(2ωt) + \sin(4ωt) ## What is the time period of the above periodic function? The following is given in my book: Period is the least interval of time after which the function repeats. Here, ##\sin(ωt)## has a period ##T_o =...

I was planning to find the value of N by taking the integral of φ*(x)φ(x)dx from -∞ to ∞ = 1. However, this wave function doesn't have a complex number so I'm not sure what φ*(x) is. I was thinking φ*(x) is exactly the same φ(x), but with x+x0 instead of x-x0. Thank you

What I wanted to do was set f(x)=##x^2##/2 - xcosx+sinx And show that f(x)>0. f'(x)=x(1+sinx) First I wanted to prove that f(x)<0 in the interval (0,∞) 0≤1+sinx≤2 And thus for all x> 0 f'(x)≥0 and therefore f(x)≥f(0)=0 And it doesn't help me much because I need to f(x)>0

A composite object made of many atoms has a large mass hence a small de Broglie wavethength...and we know that recent experiments succeeded to obtain interference patterns even for such objects (for instance the C60 molecule). Did theoretician understood how a wavefunction with such a small...

Considering Bell’s theorem and the expected correlations between entangled particles or photons. In a measurement setup e.g. Like Alain Aspect‘s with 2 entangled photons. If we could make a setup that guarantees that the measurement on both photons is done at exactly the same moment, what...

In a (reversible) Carnot cycle the entropy increase of the system during isothermal expansion at temperature TH is the same as its decrease during isothermal compression at TC. We can conclude that the entropy change of the system is zero after a complete Carnot cycle. The mentioned textbook now...

Hi, it would be really nice if someone could give me a correct answer with a little explanation to this question. Thanks.

I couldn't quite answer, so looked at the solution. I just want to ensure I am undertsanding the answer correctly. The answer is given here on page 3. Q2a: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps4_sol.pdf Am I right in concluding that...

I want to express <m(x,y,z)> over a sphere of radius R in terms of $$<\rho(x,y,z)>$$ e.g $$<m>=\frac{\int_{sphere R}m(x,y,z)dv}{\int_{sphere}dv}$$ $$<m>=\frac{\int_{sphereR}(\int \rho(x,y,z)dv)dv}{\int_{sphere R}dv}$$

What I've tried is: I have defined a function g(x)=f(x)-x^2/2. g Differentiable in the interval [0,1] As a difference of function in the interval. so -x≤g'(x)≤1-x for all x∈[0,1] than -1≤g'(x)≤0 or 0≤g'(x)≤1 . Then use the Intermediate value theorem . The problem is I am not given that f' is...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.7...

Hello everyone this was a problem on one of the exams from last year and I'm having trouble with the last point ##3## my solution for ##1## $$\frac{1}{2\pi i}\int_{|z|=r}\frac{f(z)}{z}(1+\frac{z}{2re^{i\theta}}+\frac{re^{i\theta}}{2z})dz =$$ I divided this integral into 3 different ones and...

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 1: The Real Numbers ... and in particular I am focused on Section 1.7: Continuous Functions ... I need help with clarifying Definition 1.7.1 ...Definition 1.7.1 reads as follows: My question is as...

be f Differentiable function In section [0,1] and f(0)=0, f(1)=1. so: a. f A monotonous function arises in section [0,1]. b. There is a point c∈[0,1] so that f'(c)=1. c. There is a point c∈(0,1) where f has Local max. I have to choose one correct answer.

I have no idea why my sketch is wrong.

The Fourier transform of a function is called its spectrum. The set of eigenvalues of a matrix is also called a spectrum. Why the same name? Is there some hidden connection between these two?

The double factorial, ##n!## (not to be confused with ##(n!)!##), can be defined for positive integer values like so: $$n!=n(n−2)(n−4)(n−6)...(n-a)$$ Where ##(n−a)=1## if ##n## is odd or ##(n−a)=2## if ##n## is even. Additionally, the definition of the double factorial extends such that...

The page on the Wolfram function DiracDelta has this example: Integrate[DiracDelta[x] Cos[x], {x, -Infinity, Infinity}] It's the first example under Examples > Basic Examples. When I run it in Wolfram Cloud, it says "Invalid integration variable or limit(s) ..." and shows this result: What's...

The components of the energy tensor are defined sometimes as the flux of the ith component of the momentum vector across some component jth of constant surface. But isn't the tensor a function of points of spacetime just as the metric? How can you evaluate a surface of j when the tensor is a...

Suppose I have ##y = x^2##. By inverse, I mean ##x = \pm \sqrt y##. How can I get Mathematica to do that?

Hi everyone, concerning serie representation of psi function. In te solution of bessel function of the second kind we have the following expressions for the psi functions psi(m+1) and psi(n+m+1) then they give the series for the two psi functions ie(or digamma function) sum k from 1 to m of...

Hello, I'm trying to follow Wolfram to do a least square fitting. There are multiple summations in the two equations to find the coefficients. Are the i's the same in this case? Thanks!

Where did the k come from? I was expecting the index to be t.

image due to macros in Overleaf ok I think (a) could just be done by observation by just adding up obvious areas but (b) and (c) are a litte ? sorry had to post this before the lab closes

How about I prove that to be false instead? $$z = x$$ $$z'_x(x + y) = 1$$ $$z'_y(x + y) = 0$$ $$z'_x - z'_y = 1$$

Suppose: - that I have a function ##g(t)## such that ##g(t) = \frac{dy}{dt} ##; - that ##y = y(x)## and ##x = x(t)##; - that I take the derivative of ##g## with respect to ##y##. One one hand this is ##\frac{dg}{dy} = \frac{dg}{dx}\frac{dx}{dy} = \frac{d^2 y}{dxdt}\frac{dx}{dy}##. On the other...

Hi PF! I have data that I need to interpolate (don't want to go into details, but I HAVE to interpolate it). I'm trying to find the local maximas on a given domain. I've looked everywhere and still haven't been able to do it? Seems most people work with NDSolve, but I don't use that function...

Summary:: Wave function of a laser beam before it hits the diffraction grating So I'm reading "Foundations of Quantum Mechanics" by Travis Norsen. And I've just read Section 2.4 on diffraction and interference. And he derives a lovely formula for the wave function of a particle after it leaves...

I am given the following two equations: and where E_1 is an output with corresponding input e and theta_o is an output with corresponding input E_3. The solutions that I was given are as follows: Unfortunately, I do not understand at all how to work out the block diagrams from the equations...

$$<H(\omega)>=\sum_{j} χ_{HAj}h_j(\omega)$$ Where $$χ_{HA}=\frac{1}{2\hbar} Tr{{\rho}[{H(t)},{A(0)}]}$$. But $$[H(t),A(0)]=[H_o,A(0)]-[A(t)h,A(0)]=-h_0 cos(\omega t)[A(t),A(0)]$$. So $$χ_{HA}=-\frac{1}{2\hbar}Tr(\rho h_0 cos(\omega t)[A(t),A(0)])=-h_0cos(\omega t)χ_{AA}$$. Then...

Homework Statement:: find the function of motion Homework Equations:: none i could find the amplitude and the phase angle but i can't find the phase difference and the function of motion.

So using $$L=\frac{mv^2}{2} - \frac{1}{2} m lnx$$ and throwing it into the Euler-L equation I agree with kcrick & OlderDan that we can manipulate this to either $$\frac{d}{dt} m\dot{x} = -\frac{m}{2x}$$ or $$2vdv = -\frac{dx}{x}$$ but I'm not having any epiphanies on how to turn the above into...

I am trying to create a GUI for a phyiscs project and I need subscripts of these things. `H_0`, `Omega_b`, `Omega_dm`, `Omega_\Lambda` `Omega_r` in the form of latex My code is something like this import PySimpleGUI as sg sg.change_look_and_feel('Topanga') layout = [...

Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$ ok not real sure what the answer is but I did this (could be easier I am sure} rewrite as $y=(2x+1)^3$ exchange x and rename y to g $x=(2g+1)^3$ Cube root each side...

(a) I find the geometric distribution $$X~G0(3/8)$$ and I find p to be 0.375 since the mean 0.6 = p/q. So p.g.f of X is $$(5/8)/(1-(3s/8))$$. (b) Not sure how to find the p.g.f of Y once we know there are 6 customers?

Hello all, I am trying to solve a limit: \[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\] I found many suggestions online, from complex numbers to Taylor approximations. Finally I found a reasonable solution, but one move there doesn't make sense to me. I am attaching a picture: I have marked...

I tried to calculate the Fourier transform to get the amplitude, but I got lost

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the concepts in Proposition 8.6 ...Proposition 8.6...

Hello! I want to fit a function to the curve I attached (the first image shows the full curve, while the second one is a zoom-in in the final region). Please ignore the vertical lines, what I care about is the main, central curve. It basically goes down slowly and then it has a fast rise. What...

Suppose I have an initial condition function ##f(x,t_0 )##, which is everywhere twice differentiable w.r.t. the variable ##x##, but the third or some higher derivative doesn't exist at some point ##x\in\mathbb{R}##. Then, if I evolve that function with the diffusion equation...

Hello everyone, I want to calculate the following limits: \[\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}\] using the sandwich rule, where [xa] is the integer part function defined here: Integer Part -- from Wolfram MathWorld I am not sure how to approach this. Any assistance will be...

This is a wonderful area of active research. Early in my EE career, I was interested in trying to use IC-scale nerve interfaces to bypass spinal cord injuries, but the science of interfacing electronics to nerve cells for long-term use was not developed enough. Even today, it is a problematic...

Referencing: http://www.vlsiinterviewquestions.org/2012/07/21/inverted-temperature-dependence/ Mobility decreases in a MOSFET with increasing temperature However, referencing: https://www.quora.com/Why-does-resistivity-of-semiconductors-decrease-with-increase-in-temperature Resistivity...

Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0. (a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0. (b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0. If you use a major theorem, then cite the theorem...

hey there I'm struggling on finding the domain of the following function log (xy2)+x2y) I then do xy(y+x)>0 but then i don't know what to do with xy one attempt \begin{cases} y+x>0\\ x>0\\ y>0 \end{cases} union \begin{cases} y+x<0\\ x<0\\ y<0 \end{cases} but this doesn't lead to the...

Let f1, f2: {0,1, ..., 24} → {0,1, ..., 24} be such functions that f1 (k) = k + 1 for k <24, f2 (k) = k for k <24 and f1 (24) = f2 (24) = 0. Let gi1, i2, ..., I am (k) = fi1 (fi2 (... fim (k) ...)) for i1, i2, ..., im∈ {1,2}. Find the largest m for which irrespective of the selection i1, i2...