Functions Definition and 1000 Threads

Write $\cot^2(x)-\csc^2(x)$ In terms of sine and cosine and simplify So then $\dfrac{\cos ^2(x)}{\sin^2(x)} -\dfrac{1}{\sin^2(x)} =\dfrac{\cos^2(x)-1}{\sin^2(x)} =\dfrac{\sin^2(x)}{\sin^2(x)}=1$ Really this shrank to 1 Ok did these on cell so...

$\tiny{from\, steward\, v8\, 6.4.2}$ find y' $\quad y= x\ln{x}-x$ so $\quad y'=(x\ln{x})'-(x)'$ product rule $\quad (x\ln{x})'=x\cdot\dfrac{1}{x}+\ln{x}\cdot(1)=1+\ln{x}$ and $\quad (-x)'=-1$ finally $\quad \ln{x}+1-1=\ln{x}$...

Hi, I'm a bit confused about Bloch functions. This is what, I think, I understood: can someone please tell me what's wrong? From Bloch's theorem we know that the wave-function of an electron inside a periodical lattice can be written as ##ψ_k(r)=u_k(r)e^{ik⋅r}##. We hope that far from a lattice...

Hello everyone, I have noticed a striking similarity between expressions for creation/annihilation operators in terms position and momentum operators and trigonometric expressions in terms of exponentials. In the treatment by T. Lancaster and S. Blundell, "Quantum Field Theory for the Gifted...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows: In the above proof by...

I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ... I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ... The statement of Theorem 3.1.1 (A1) reads...

Can anyone explain the logic behind the answer? Taken from HiSet free practice test

Suppose I have Gaussian white noise, with the usual dirac-delta autocorrelation function, <F1(t1)F2(t2)> = s2*d(t1-t2)*D12 Where s is the standard deviation of the Gaussian, little d is the delta function, and big D is the kronecker delta. For concreteness and to keep track of units, say F...

I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me? ##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\ I(R_j) =...

I found three independent functions as solutions for this equation d/dr(r^2dR/dr) = 6R (cauchy equation) r^2 , r^(-3) , (1/7)r^6. But i read that a second order linear differential eqn has only two independent solutions. Why this happened?

I know it is something simple that I am missing, but for the life of me I am stuck. So, I used the identity ##sin(a)sin(b)+cos(a)cos(b)=cos(a-b)## which gives me $$\int^{\infty}_{-\infty}dx\:f(x)\delta(x-y)=\int^{\infty}_{-\infty}dx\:f(x)\frac{1}{2L}\sum^{\infty}_{n=-\infty}\lbrace...

Two KL functions $f_1:\mathbb{R}^n\rightarrow \mathbb{R}$ and $f_2:\mathbb{R}^n\rightarrow \mathbb{R}$ are given which have KL exponent $\alpha_1$ and $\alpha_2$. What is the KL exponent of $f_1+f_2$?

I'm interested in the following problem: given a random number n (n can be gigantic), how do we find a pair function+input(s) whose output is n such that the input(s) are relatively small in size? This problems arises in data compression; consider the bits that make up a file (or a substring of...

I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...

Problem Statement: I am trying to understand how to compute the surface an N-sphere , for large N, to leading order (and exactly) Given a vector J with norm N, with N large, how does one compute the volume integral ? That is, what representation of the delta function. And what is the exact...

I tried using Kirchhof's current law, and to pose the problem in matrix form as ##\frac{dv}{dt}=Mv## with## v## the vector of the ##3## potentials at nodes ##1, 2## and ##3##, and ##M## is a ##3x3## matrix. it would be enough to show me which will be the differential equations, I would proceed...

1] Let X,Y,Z be independent, identically distributed random variables, each with density $f(x)=6x^5$ for $0\leq x\leq 1,$ and 0 elsewhere. How to find the distributon and density functions of the maximum of X,Y,Z.2]Let X and Y be independent random variables, each with density $e^{-x},x\geq...

Let F(x,y) be the joint distribution for random variables X and Y (not necessarily independent). Is ##lim_{y\to \infty}F(x,y)## the distribution function for X? I believe it is. How to prove it?

If it is the asymptotic behavior of the Airy's function what it's used instead of the function itself: Does it mean that the wkb method is only valid for potentials where the regions where ##E<V## and ##E>V## are "wide"?

Moved from technical forum, so no template is shown Summary: I have the expression sin(2x) + sin(2[x + π/3]) and I have to write this in terms of a single function (a single harmonic, rather saying). But I don't know how to do this, and... it seems a little bit weird for me, because I'm merging...

## ## Well I start with equation 1): ## e^{b\theta }=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in\theta } ## If ## \theta =0 ## ##e^{b(0)}=\frac{sinh(b\pi )}{\pi }\sum_{-\infty }^{\infty }\frac{(-1)^{n}}{b-in}e^{in(0) }## ##1=\frac{sinh(b\pi )}{\pi...

Hi, I'm confused between H(ω) and H(s) as transfer functions. The textbook defines both as transfer functions though the term transfer function is mostly reserved for H(s) as far as I can tell. I have read that poles and zeroes of H(s) are helpful in determining the stability. Are poles and...

If averaging of a function over a volume is defined as ##\frac{\int_v f(x,y,z,t) dv}{\int_v dv}##. Now if the average ##f^2(x,y,z,t)## is given 0 over a volume,then ##f(x,y,z,t)## has to be necessarily 0 in the volume domain??

The definition of the Riemann sums: https://en.wikipedia.org/wiki/Riemann_sum I'm stuck with a problem in my textbook involving upper and lower Riemann sums. The first question in the problem asks whether, given a function ##f## defined on ##[a,b]##, the upper and lower Riemann sums for ##f##...

Good evening! Going through a bunch of calculations in Ashcroft's and Mermin's Solid State Physics, I have come across either an error on their part or a missunderstanding on my part. Suppose we have a concatenated function, say the fermi function ##f(\epsilon)## that goes from R to R. We know...

Hi all, What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical...

So, I found these statements and I need your assistance to prove them since my body condition is not fit enough to think that much. 1. The quadratic equation whose roots are k less than the roots of ax^2+bx+c=0 is a(x+k)^2+b(x+k)+c=0. 2. The quadratic equation whose roots are k more than the...

Hello All, I have a question regarding a MATLAB homework problem. I am learning about logical functions and selection structures. Here is the question: The height of a rocket (in meters) can be represented by the following equation: height=(2.13*t^2)-(0.0013*t^4)+((0.000034*t^(4.751)) create a...

Hello, I would like to write a product of two wave functions with a single ket. Although it looks simple, I do not remember seeing this in any textbook on quantum mechanics. Assume we have the following: ##\chi(x) = \psi(x)\phi(x) = \langle x | \psi \rangle \langle x | \phi \rangle## I would...

I really have no clue how to start this. I think I might have to use Pythagoras but I'm really not sure.

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand the vector space of alternating multilinear functions ... The relevant text from Shifrin reads as follows: In...

I am working through Tu's "An Introduction to Manifolds" and am trying to get an understanding of things with some simple examples. The definitions usually seem simple and understandable, but I want to make sure I can use them for an actual function. I've worked a few problems below that my...

Homework Statement I have encountered this problem from the book The Physics of Waves and in the end of chapter six, it asks me to prove the following identity as part of the operation to prove that as the limit of ##W## tends to infinity, the series becomes an integral. The series involved is...

Hi PF Given some linear differential operator ##L##, I'm trying to solve the eigenvalue problem ##L(u) = \lambda u##. Given basis functions, call them ##\phi_i##, I use a variational procedure and the Ritz method to approximate ##\lambda## via the associated weak formulation $$\langle...

Homework Statement On ##L_2[0,2\pi]## where ##e = \{ 1/\sqrt{2 \pi},1/\sqrt{\pi}\sin x,1/\sqrt{2 \pi}\cos x \}##. Given ##f(x) = x##, find ##Pr_e f##. Homework Equations See solution. The Attempt at a Solution I take $$e \cdot \int_0^{2\pi} e f(x) \, dx = \pi - 2 \sin x.$$ Look correct?

Is the following true? ψ*∇^2 ψ = ∇ψ*⋅∇ψ It seems like it should be since you can change the direction of operators.

Gravity acting on a 10 kg,mass produces a force of $F_g=\langle 0, -98\rangle$ Newtons. If the mass is suspended from 2 wires which both form $30^\circ$ angles with the horizontal, then what forces of tension are required in order for the mass to hang motionless over time? Answer. I computed...

Homework Statement If a staight ε: y=(-λ+μ)x +2λ -μ , (where μ and λ are real numbers) passes through point A(0,1) and is parallel to an other straight lin. ζ: y= -2x + 2008 find λ and μ Homework EquationsThe Attempt at a Solution It is clear that when x=0 we know that 2λ-μ=1 which is one of...

What do the average velocities on the very short time intervals [2,2.01] and [1.99,2] approximate? What relationship does this suggest exist between a velocity on an interval [a,b] and a velocity near t=a+b/2 for this type of polynomial?

Hi all, I am slightly confused with regard to some ideas related to the GCE and CE. Assistance is greatly appreciated. Since the GCE's partition function is different from that of the CE's, are all state variables that are derived from the their respective partition functions still equal in...

Let X1, · · · , Xn be a simple random sample from some finite population of values {x1, · · · xN }. Is the estimate \frac{1}{n} \sum_{i}^{n} f(Xi) always unbiased for \frac{1}{N} \sum_{i}^{N} f(xi) no matter what f is?My thinking: I don't think all f's are unbiased, because not all sample...

Homework Statement Find the functions: f: (0, ∞) → ℝ and g: ℝ → ( 0, ∞) such that f o g = Iℝ (Iℝ denotes identity function on ℝ). Homework EquationsThe Attempt at a Solution I am having trouble working backwards. I know that (f o g)(x) is f(g(x)). I am unsure if this is correct but would f o...

I was reading Griffith's introduction to QM book and he finds the time independent Schrodinger equation by assuming the wave function to be the product of two independent functions. He eventually gets to this: ih(∂ψ/∂x)/(ψ) = -(h^2/2m)*(∂''φ/∂x^2)/φ + V he says that "the...

Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...

Homework Statement Given: Ψ and Φ are orthonormal find (Ψ + Φ)^2 Homework Equations None The Attempt at a Solution Since they are orthonormal functions then can i do this? (Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?

1)* What are sine and cosine functions called in relation to Pi? 2) What is the exponential function called in relation to cosine and sine functions? 3) What are the other smooth, continual nested (or iterative) root functions (that are similar to sine and cosine) called in relation to...

I do not know to start. Here is the problem.Determine if the given function is piecewise continuous, piecewise smooth, or neither. Here $x\neq0$ is in the interval $[-1,1]$ and $f(0)=0$ in all cases. 1. $f(x)=sin(\frac{1}{x})$ 2. $f(x)=xsin(\frac{1}{x})$ 3. $f(x)={x}^{2}sin(\frac{1}{x})$ 4...

Dear Everyone, I do not know how to begin with the following problem:Suppose that $f$ is $2\pi$-periodic and let $a$ be a fixed real number. Define $F(x)=\int_{a}^{x} f(t)dt$, for all $x$ . Show that $F$ is $2\pi$-periodic if and only if $\int_{0}^{2\pi}f(t)dt=0$. Thanks, Cbarker1

I am trying to understand why maxwell equations are correct in any reference frames? While i started to understand of his laws of physics a bit i could not imagine why he uses hyperbolic functions such as coshw instead of spherical ones in position and time relation between moving frames...