Function Definition and 1000 Threads

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...

$$<H(\omega)>=\sum_{j} χ_{HAj}h_j(\omega)$$ 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...

1.) Laplace transform of differential equation, where L is the Laplace transform of y: s2L - sy(0) - y'(0) + 9L = -3e-πs/2 = s2L - s+ 9L = -3e-πs/2 2.) Solve for L L = (-3e-πs/2 + s) / (s2 + 9) 3.) Solve for y by performing the inverse Laplace on L Decompose L into 2 parts: L =...

First some background, then the actual question... Background: (a) Very simple example: if we take ##Asin(x+ϕ)+0.1##, the average is obviously 0.1, which we can express as the integral over one period of the sine function. (assume that we know the period, but don't know the phase or other...

As promised, here is the original question, with the integral written in a more legible form.

Here's the circuit in question: Solution: Now, when I try simulating in LTSpice, this is what I get: So, Vout appears to be around -13 V, which doesn't agree with the equation if V1=V2= 5 is plugged in. Does anyone see the mistake here? THanks.

I've been playing around with Up-Arrow notation quite a lot lately and have come up with the following "thought experiment" so to speak. Consider the following function: $$f(x)=(−ln(x↑↑(2k)))↑↑(2k+1)$$ $$\text{Where }k∈\mathbb{Z} ^+$$ In the image below we can see some examples of what this...

The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.

Suppose we have a function which looks like this: It seems like it meets criteria of probability density functions: this function is asymptotic to zero as x approaches infinity and also it is not negative. My question is: if at some points this function reaches zero (as I have shown above)...

There is an arbitrarily complicated function F(x,y,z). I want to find a simpler surface function G(x,y,z) which approximates F(x,y,z) within a region close to the point (x0,y0,z0). Can I write a second-order accurate equation for G if I know F(x0,y0,z0) and can compute the derivatives at the...

Firstly, this is not a homework question. I found a worksheet online with an example of a square law circuit built using log-antilog operational amplifiers. I tried to derive the transfer function but I can't seem to eliminate the reverse saturation current term ##I_S##. I would really...

Here is the code that I wrote: import numpy as np global m, n, p, q, arr1, arr2def input(): # Input for first matrix: print("Enter the number of rows of the first matrix: ", end="") globals()['m'] = int(input()) print("Enter the number of columns of the first matrix: ", end="")...

I have ##3x^{2/3}## as an even function although there is some debate as to this in another thread I started but the (5-x) factor means the function is neither odd or even. I also see the domain as all real numbers. Hopefully this is right ? To find the critical points I differentiate f(x) to...

My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I have yet another question regarding Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...

Consider, for example, the gluon propagator $$D^{\mu\nu}(q)=-\frac{i}{q^2+i\epsilon}[D(q^2)T^{\mu\nu}_q+\xi L^{\mu\nu}_q]$$ What exactly is the renormalized gluon dressing function ##D(q^2)## and what is its definition? My interest is in knowing if I can then write the bare version of this...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with an aspect of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III, reads as...

so ergo the function is neither convex nor concave. but graphing it in a machine it looks convex...

Hi, in some standards such as JESD204B or DVB-S2 a so called scrambler function is defined. As far as I understand this scrambler is a means of spreading spectrum but in data link layer. Is it correct? Senmeis