Functions Definition and 1000 Threads

I am trying to show that that the Airy functions defined below satisfy: $W[Ai(x),Bi(x)]=1/\pi$. $$Ai(x)=\frac{1}{\pi} \int_0^\infty \cos(t^3/3+xt)dt$$ $$Bi(x)=\frac{1}{\pi}\int_0^\infty \bigg[ \exp(-t^3/3+xt)+\sin(t^3/3+xt)\bigg]dt $$ I tried to compute it directly but I got stuck, here's the...

I am in the trigonometry section of my precalculus textbook by David Cohen. In Section 6.2, David explains how to evaluate trig functions without using a calculator but it is not clear to me. Sample: Is cos 3 positive or negative? How do I determine if cos 3 is positive or negative without...

where rk are the roots of , Find S. -------------- I don't have a decent approach. I am very bad at summations. Although, my friend asked me to assume complex analysis and I still couldn't do it. A simple hint to this would be appreciated. Peace.

Define the numbers $e_k$ by $e_0 = 0$, $e_k = \exp(e_{k-1})$ for $k \geq 1$. Determine the functions, $f_k$, for which \[f_0(x) = x, \;\;\;\;f’_k = \frac{1}{f_{k-1}f_{k-2}\cdot\cdot\cdot f_0}\;\;\;\; for\;\;\; k \geq 1.\] on the interval $[e_k, \infty)$, and all $f_k(e_k) = 0.$

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with the proof of Proposition 2.2.9 ... ... Duistermaat and Kolk's Proposition 2.2.9 read as follows: In the above text...

Differentialbility & Continuity of Multivariable Vector-Valued Functions ... D&K Lemma 2.2.7 ... I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Lemma 2.2.7 (Hadamard...) ... ... Duistermaat and Kolk's Lemma 2.2.7 and its proof read as...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 4 on page 66 ... Kantorovitz's Example 4 on page 66 reads as follows:In the above example, Kantorovitz...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 4 on page 66 ... Kantorovitz's Example 4 on page 66 reads as follows: In the above example, Kantorovitz...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 3 on pages 65-66 ... Kantorovitz's Example 3 on pages 65-66 reads as follows...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 3 on pages 65-66 ... Kantorovitz's Example 3 on pages 65-66 reads as follows: In the above example, we read...

I have two functions ##\phi(t)=\cos(\omega t)## and ##f(t)=u(t)−u(t−k)## with ##f(t)=f(t+T)##, ##u(t)## is the unit step function. The problem is to find Laplace transform of ##\phi(t) \cdot f(t)##. I have tried convolution in frequency domain, but unable to solve it because of gamma functions...

Homework Statement why this formula works ? Homework EquationsThe Attempt at a Solution when i take the derivative of the right side ,,, there is an additional "a" in the numerator in place of 1,, why the derivative of arcsine of (u/a) not exactly same with the expression under the integral sign

Just a general question here. So for a polynomial function, the behavior of the graph at the zeros is determined by the evenness or oddness of the magnitude of the zeros. If the magnitude is odd, the graph will cross the zero. If the magnitude is even, it will bounce at the zero. Why is this...

Hi everyone. So I was studying and they say there is a force function (and a potential energy, I suppose) for every fundamental interaction. So, they always show the gravitational and electromagnetic force/potential energy functions for these, and they always mention the other two (plus the...

Homework Statement Graph ##y=tan\left(x-\frac {π}{4}\right)## Homework Equations N/A The Attempt at a Solution To graph a tangent function, I first find the vertical asymptotes to set the boundaries for the graph: To do so, set what's inside the parentheses equal to ##\frac π 2## and ##-\frac...

(I) Find the limit (x,y)->(0,0) of F, then prove it by definition. (II) Find the limit and prove it by definition of: as (x,y) approach (C,0), C different from zero. I have previously asked it on Quora, but it doesn't appear to have answers any...

Hi! (Not sure which forum to pick for this question. This looked like the best one. I apologies if it is not) I have a number of functions (say m functions) with integer domain. All functions are increasing. (Increasing in the sense not decreasing, $f(n+1) \ge f(n)$.) I want an algorithm to...

Hi, in a text provided by DrDu which I am still reading, it is given that "the momentum operator P is not self-adjoint even if its adjoint ##P^{\dagger}=-\hbar D## has the same formal expression, but it acts on a different space of functions." Regarding the two main operators, X and D, each has...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.8 ... Duistermaat and Kolk"s Example 1.3.8 reads as follows:In the above example we read the...

Suppose we have a particle, let's say an electron, in a box of size D. And we add another one next to it at some distance L center to center. How do we solve for the wavefunctions of the electron. Can it be solved in normal QM or do we need QFT. Thanks.

Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...

<Moderator's note: This is a spin-off from another thread.> I will find out axioms to find out an answer to a question - axioms guarantees that my solution to a mathematical problem is correct. I have another question: A function 'y' in 'x' yields a single value as output on an input. Is there...

Homework Statement This is for College Algebra. Describe what is happening to the graph of the function ##f\left(x\right)=\sqrt {1 - x}+2##. Homework Equations N/A The Attempt at a Solution This can be rewritten as ##f\left(x\right)=\sqrt {-x + 1}+2## My conundrum: Both my text, as well as...

Hi PF! Given an autonomous function ##f(x,j)##, I am trying to create a ##j##-vector so that I can "dot" this into a ##j \times 1## vector. For example, if I have ##f = @(x,j) \sin x j## I would like to create something like ##v = [\sin x,\sin 2x, \sin 3x]## so I could dot this into a vector...

Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?

Hi, I have a strange nonlinear operator which yields non-Hermitian solutions when treated in a simple ODE, ##H\Psi##=0. It appears from a paper by Dr Du in a different posting, that an operator can be non-self-adjoint in one domain, but be self-adjoint in another domain defined by the interval...

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Lemma 1.3.3 ... Duistermaat and Kolk"s proof of Lemma 1.3.3. reads as follows:In the above proof...

Hi, in QM literature the inadmissible solutions to the Schrödinger eqn are often , if not always, quoted in the text as "inadmissible", because they are discontinuous, not-single valued, not square integrable and not infinitely differentiable. However in a discussion with Dr Du yesterday...

Dear friends, I am new to this forum and I need urgent help in solving these two questions as I am due to submit them tomorrow early morning. Please help me in solving these two questions. Waiting for urgent response. 1: A farm’s profit is given by π = 100x + 80y + 2xy− x(square) − 2y(square) −...

Since the distances from the origin $\displaystyle \begin{align*} \rho \end{align*}$ are the same, we can say $\displaystyle \begin{align*} \rho = \frac{3\,\alpha}{2} \end{align*}$ and $\displaystyle \begin{align*} \rho = \beta + \pi \end{align*}$, giving $\displaystyle \begin{align*}...

Recently we are studying the physical model using the confluent Heun function, who konws how to solve it numerically by software. We use Maple but it seems not to work well. This means that the results obtained are not correct...

Not a particular problem to wonder about but more of a general question, when one has a free body diagram, when is it best to use sine and when is it best to use cosine? I am reviewing some of my tests for a final, and having previously re-read my forces chapter, I thought that angles rising...

Hi, I was studying for my final exam on statistical physics and a doubt raised on my head that was truly strong and disturbing (at least, for me), and that I couldn't answer to myself by now. The doubt is: Given that we have in d dimensions a fermion non interacting gas, the statistical...

Hey guys, I have a couple of questions here. One, I was just wondering if someone could elaborate on, and the second, I worked it out, but more by guessing. I was hoping someone would be able to help explain both. Here is the first of the two questions So, part a was fairly straightforward...

Find all functions, $f: \mathbb{N}\rightarrow \mathbb{N}$, satisfying \[f(mn)=f(m)f(n),\: \: \: and \: \: \: m+n \: \: |\: \: f(m)+f(n)\]for all $m,n \in \mathbb{N}$.

Homework Statement For a sequence ##\{f_n\}## of measurable functions with common domain ##E##, show that the following functions are measurable: ##\inf \{f_n\}##, ##\sup \{f_n\}##, ##\lim \inf \{f_n\}##, and ##\lim \sup \{f_n\}## Homework EquationsThe Attempt at a Solution It suffices to...

I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 2: Relations, Functions and Orderings; and, in particular on Section 3: Functions I need some help with H&J's depiction of...

Studying for my complex analysis final. I think this should be a simple question but wanted some clarification. "Extend the formula $$\frac{1}{2i\pi} \int_\omega \frac{h'(z)}{h(z)}\, dz = \sum_{j=1}^N n_j - \sum_{k=1}^M m_k$$ to prove the following. Let $g$ be analytic on a domain...

1. Homework Statement I've been using a recurrence relation from "Adv. in Physics"1966 Nr.57 Vol 15 . The relation is : where Rnl are radial harmonic oscillator wave functions of form: The problem is that I can't prove the relation above with the form of Rnl given by the author(above). I've...

Homework Statement Hi all, I came across these steps in my notes, relating to a step whereby, $$\hat{G} (k, t - t') = \int_{-\infty}^{\infty} e^{-ik(x - x')}G(x-x' , t-t')dx$$ and performing the following operation on ##\hat{G}## gives the following expression, $$[\frac{\partial}{\partial t}...

I am so sorry for having posted this challenge/puzzle with a serious typo: The roots of the equation should be functions of $a, d$ and $e$. In my old version I wrote $a, b$ and $e$. I will see to, that future challenges are properly debugged before posting.For $e \ne 0$, determine the roots...

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 8.2 Differentials ... I need help in fully understanding Browder's comments on Definition 8.9 ... ... Definition...

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 8.2 Differentials ... I need help in proving Proposition 8.12 ... ... Proposition 8.12 and the definitions...

Hi, Apologies if this questions is really easy but it is something quite subtle which is annoying me. In my book of quantum physics it gives a wave function of definite momentum: ψ = Aeipx/ħ It goes on to say that since there is a momentum 'p' in the exponential then the momentum is known...

Homework Statement The piece wise function is -x-2, x<-1 x^2-3x, -1≤ x ≤5 3x+5, x>5 The problem is to transform the function with...

1. The problem statement, all variables, and given/known data I am given a distribution function f(x) that tells me the number of objects with a certain physical property x (such as having a certain mass or temperature) and I need to calculate the total number of objects, the average value of...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding another aspect of the proof of Proposition 1.3...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding aspects of Proposition 1.3 ...Proposition 1.3 and its...

Hi PF! When using Mathematica I input the code f1[a_, n_] := f1[a, n] = Join[Table[LegendreP[k, x], {k, 0, n, 1}], Table[LegendreP[k, x], {k, 1, n, 1}]] Then when I type ##f1[1, 3] ## I get an output. I then change the second table of ##f1## to start at 0 instead of 1, recompute...