Functions Definition and 1000 Threads

How come a+a- ψn = nψn ? This is eq. 2.65 of Griffith, Introduction to Quantum Mechanics, 2e. I followed the previous operation from the following analysis but I cannot get anywhere with this statement. Kindly help me with it. Thank you for your time.

Homework Statement [/B] In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ? 2. Homework Equations The Attempt at a Solution

I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?

Functions are pretty simple things , they just express a relationship between two different quantities How do i express this function in terms of y(x) = something ?

hi, i was thinking that every function that satisfies the conditions $$f(0)=1$$ $$f(n+1)=(n+1)f(n)$$ could be a generalization of the factorial function, and why the gamma function is the only function that complies with this conditions? I mean why don't exist other functions, or functions...

Hello, In C (or C++), a function is a body of instructions. Functions can be classified as functions that 1) receive inputs and produce outputs 2) receive no inputs and produce no outputs 3) receive inputs and produce no outputs 4) receive no inputs and produce outputs For case 1) and 4), the...

Hello everyone. So I have a test coming up and I am struggling with the concept of figure out what the initial phase or angle of a transfer function is. For instance, consider the following transfer function: L(s) = 4/s(.4s+1)(s+2) So the initial angle for L(s) is -90 degrees. Is there a...

Let f be a function with the intermediate value property. In addition, let it have the property that |f(x)-x_n|\le M\cdot sup_{n,m}|f(x_n)-f(x_m)|, where M is a constant and x_n is a sequence converging to x. Then, can we show that f is continuous? I think we have to tackle this problem by...

I seem to have forgotten where I have seen these particular rules; I need them for my research. I think I saw them in a book somewhere... but that was around 4 years ago. Does anyone know where can I find the rules stipulated by statistical mechanics/thermodynamics on the alpha function α(TR)...

Gauge symmetry is not a symmetry. It is a fake, a redundancy introduced by hand to help us keep track of massless particles in quantum field theory. All physical predictions must be gauge-independent...

Homework Statement See attached. The solution of part e) is ##C=4\psi(a)## I am looking at part e, the answer to part d being that the principal parts around the poles ##z=0## and ##z=-a## are the same. Homework EquationsThe Attempt at a Solution [/B] Since we already know the negative...

Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...

Homework Statement [/B] please see attached. b) The solution seems a bit vague is the idea here, what this comment is saying, that since this is a simple zero the form of ##lim_{z\to a} f_a(z) (z-a)=0## since, crudely, it is of the form ##\frac{0.0}{0}##. Compared to the point ##z=-a##...

Homework Statement I am trying to show that the integrator is unstable by giving examples of bounded inputs which produce unbounded outputs (i.e. a bounded function whose integral is unbounded). Note: The integrator is a system which gives an output equal to the anti-derivative of its input...

Homework Statement [/B] Let X be a random variable with support on the positive integers (1, 2, 3, . . .) and PMF f(x) = C2 ^(-x) . (a) For what value(s) of C is f a valid PMF? (b) Show that the moment generating function of X is m(t) = Ce^t/(2− e^t) , and determine the interval for t for...

I need help in understanding how Jacobian Elliptic Functions are interpreted as inverses of Elliptic Functions. Please reference the wiki page on Jacobian Elliptic functions: https://en.wikipedia.org/wiki/Jacobi_elliptic_functions For example, if $$u=u(φ,m)$$ is defined as $$u(φ,m) =...

Homework Statement Hi, I am trying to understand the attached: I know that if two functions have zeros and poles at the same point and of the same order then they differ only by a multiplicative constant, so that is fine, as both have a double zero at ##z=w_j/2## and a double pole at...

Homework Statement I have that ##(\psi(z)-e_j)^{1/2}=e^{\frac{-n_jz}{2}}\frac{\sigma(z+\frac{w_j}{2})}{\sigma(\frac{w_j}{2})\sigma(z)}## has period ##w_i## if ##i=j## and period ##2w_i## if ##i\neq j## where ##i,j=1,2,3## and ##w_3=w_1+w_2## (*) where ##e_j=\psi(\frac{w_j}{2})## I have...

I'm having issues with the first four questions and have uploaded them. My attempts are shown below. 1. a) True, all elements of E are even b) False, 0 is not a multiple of 3 c) True, 8 is even and 9 is a multiple of 3 d) No idea e) False, 6 is an element of E and T f) No idea 2. a) You can...

I want to figure out whether the functions are differentiable at c. I think I should use some of the trig identities, but I'm not sure which ones. Any tips?

Homework Statement [/B] Homework Equations Provided in (1). The Attempt at a Solution I think (a) is no because, though ##c_1g > f,## the actual un-vertically-translated ##g## could be less than ##f,## meaning its lower bound ##c_2h < f## over ##c_2 \geq 1,## meaning ##h < f.## Am I...

Does anyone know how I can prove the following equation? ##\displaystyle \frac 1 {d-1}-\int_0^1 \frac{dy}{y^d} \left( \frac 1 {\sqrt{1-y^{2d}} }-1\right)=-\frac{\sqrt \pi \ \Gamma(\frac{1-d}{2d})}{2d \ \Gamma(\frac 1 {2d})} ## Thanks

I have a question stating to derive the functions x |-> f_1(x)=x^3 and f_2(x)=thirdrootof(x) on their domains of definition based on the asymptotic relative condition number KR = KR(f,x). I'm not sure where to start with this question, I'm not sure if I even understand it. Do I find the...

The problem statements, all variables and given/known data: Question 1 Question 2 Relevant equations: Provided in question snips. The attempt at a solution: Question 1: I think qux is the answer because it properly increments k by 1 in each iteration. j = j * 2 means j is always 0 so its...

Hi, beginner coder here. I have a somewhat solid understanding of both vectors and functions, and have used the two of them many times, but I'm have trouble coding functions that have vectors in their parameters and as their return values. Another thing I'm having trouble with is calling the...

Just wondering if anyone could help me get in the right direction with these questions and/or point me to some material that will help me better understand how to approach these questions In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R. (a) Show that a function...

i want to know if any real function can be expressed as: f(x)=g(x)+h(x) such as g(x) is an increasing function and h(x) is a decreasing function? thanks

Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf). $$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\...

Homework Statement ##\Omega = {nw_1+mw_2| m,n \in Z} ## ##z_1 ~ z_2 ## is defined by if ##z_1-z_2 \in \Omega ## My notes say ##z + \Omega## are the cosets/ equivalence classes , denoted by ##[z] = {z+mw_1+nw_2} ## Homework Equations above The Attempt at a Solution So equivalance classe...

Hi, Given the algebraic function ##w(z)## defined implicitly as ##f(z,w)=a_0(z)+a_1(z)w+a_2(z)w^2+\cdots+a_n(z)w^n=0##, is there any on-line table of genus for them? Haven't been able to find anything. I am writing some code and would like to check it against a standard source. For example...

Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails. Given: f (x, y) = (x - 1)^2 (y + 4)^2 I found the partial derivative for x and y to be the following: f_x = 2 (x - 1)(y + y)^2 f_y = 2 (y + 4)(x - 1)^2 I solved for x...

Homework Statement Where are the following functions discontinuous? f(x) = (x+2)/√((x+2)x) Homework EquationsThe Attempt at a Solution f(x) = (x+2)/√((x+2)x) = (x+2)/x√(2x) multiply both denominator and numerator by √(2x) = (x√2+2√x)/(x(2x)) Can I leave it like this and state that x ≠ 0, or...

Suppose I have some observables \alpha, \beta, \gamma whose central values and uncertainties \sigma_{\alpha}, \sigma_{\beta}, \sigma_{\gamma} are known. Define a function f(\alpha, \beta, \gamma) which has both real and complex parts. How do I do standard error propagation when imaginary...

Homework Statement Consider the set V + {all periodic *complex* functions of time t with period 1} Draw two example functions that belong to V. Show that if f(t) and g(t) are members of V then so is f(t) + g(t)Homework EquationsThe Attempt at a Solution f(t) = e(i*w0*t)) g(t) =e(i*w0*t...

The average angle made by a curve ##f(x)## between ##x=a## and ##x=b## is: $$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$ I don't think there should be any questions on that. Since ##f'(x)## is the value of ##\tan{\theta}## at every point, so ##tan^{-1}{(f'(x))}##, should be the angle made by...

Hi, I've got this: $$\sin{(A*B)}\approx \frac{Si(B^2)-Si(A^2)}{2(\ln{B}-ln{A})}$$, whenever the RHS is defined and B is close to A ( I don't know how close). Here ##Si(x)## is the integral of ##\frac{\sin{x}}{x}## But, to check it, I need to evaluate the ##Si(x)## function. I'm new with Taylor...

Hi, I got these: $$log(a+b)\approx \frac{b*logb-a*loga}{b-a} + log2 -1$$ $$tan^{-1}(a+b)\approx \frac{b*tan^{-1}2b-a*tan^{-1}2a+\frac{1}{4}*ln\frac{1+4a^2}{1+4b^2}}{b-a}$$ $$sin^{-1}(a+b)\approx \frac{b*sin^{-1}2b-a*sin^{-1}2a+\frac{1}{2}*(\sqrt{1-4b^2}-\sqrt{1-4a^2}}{b-a}$$ And, similarly for...

Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...

Hello all, I am trying to take the inversion of this function that is in Laplace domain. I've tried using a wolfram alpha solver, and I know I can probably use stehfest algorithm to numerically solve it but wanted to know if there was an exact solution. the function is...

Homework Statement How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity? y=t^2 y=1+i*t^2[/B] y=(2+3*i)/t The Attempt at a Solution I thought: y=t^2 - along a part of a line that does not pass through the...

Homework Statement See attached picture.Homework EquationsThe Attempt at a Solution At the moment, I am dealing with part (a). What I am perplexed by is the ordering of the parts. If the subbasis in part (b) does indeed generate this coarsest topology, why wouldn't showing this be included in...

Hi, I am having some trouble understanding exactly when a modified green's function is needed. Here is the general problem: Lu = (p(x)u'(x))' + q(x)u(x) = f(x), x_0 \leq x \leq x_1, p(x) > 0,\\ \alpha_0 u(0) + \beta_0 u'(0) = 0, \alpha_1 u(1) + \beta_1 u'(1) = 0 In my notes it says...

Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$ f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) , $$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and...

Homework Statement Hi I have questions on the attached lemma and proof. ##f(z)## is an elliptic function here, and non-consant ##\Omega## is a period lattice. So the idea behind the proof is this is a contradiction because the function was assumed to be non-constant but by the theorem that...

I was thinking about extending the definition of superlogarithms. I think maybe that problem can be solved if we find a function ##f## such that ##fof(x)=log_ax##. Is there some way to find such a function? Maybe the taylor series could be of some help. Or is there some method to find a...

so, continuous signals as sums of weighted delta functions can be represented like this: if you switch order of some variables you get ∫x(τ)δ(-τ+t)dτ, and since,I presume, Dirac delta "function" is even I can write it like this ∫x(τ)δ(-(-τ+t))dτ=∫x(τ)δ(τ-t)dτ=x(t) and we got ourselves a...

example code from python-course.eu def factorial(n): print("factorial has been called with n = " + str(n)) if n == 1: return 1 else: res = n * factorial(n-1) print("intermediate result for ", n, " * factorial(" ,n-1, "): ",res) return res...

Hi, I have the following: Let ##\Omega ## be a discrete subgroup of ##C##, the complex plane. If: i) ##\Omega = \{nw_1 | n \in Z\} ##, then ##\Omega ## is isomorphic to ##Z##. ii) ##\Omega = \{nw_1 + mw_2 | m,n \in Z\} ## where ##w_1/w_2 \notin R ## , then ##\Omega## is isomorphic to ##Z## x...

Is the set of all differentiable functions ƒ:ℝ→ℝ such that ƒ'(0)=0 is a vector space over ℝ? I was given the answer yes by someone who is better at math than me and he tried to explain it to me, but I don't understand. I am having difficulty trying to conceptualize this idea of vector spaces...

Homework Statement Give the example and show your understanding: [1][/B].Lets define some property of a point of the function: 1. Point is a stationary point 2. Point is a max/min of a function 3. Point is a turning point of a function If possible name a function whose point has properties of...