Functions Definition and 1000 Threads

Hey everyone . So I've started reading in depth Fourier transforms , trying to understand what they really are(i was familiar with them,but as a tool mostly) . The connection of FT and linear algebra is the least mind blowing for me 🤯! It really changed the way I'm thinking ! So i was...

I just submitted a solution for a leetcode problem, displayed below. My solution is broken into three separate functions as shown below. def Checker(dig, number): """ Purpose: Function check if the given number is divisible by each digit in the given number. Parameters...

I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function I know that to solve it we must first find...

So just based on the cauchy riemann theorem, I think: Ux = 2 = Vy = 2xy, so f(z) is differentiable on xy = 1, and also that Vx = y^2 = -Uy = 0. That doesn't make sense to me because if 0 = y^2, then y = 0, yet that wouldn't satisfy xy = 1, would it? Furthermore, I'm not sure how I would...

Summary:: I attach a picture of the given problem below, just before my attempt to solve it. We are required to show that ##\alpha_1 \varphi_1(t) + \alpha_2 \varphi_2(t) = 0## for some ##\alpha_1, \alpha_2 \in \mathbb{R}## is only possible when both ##\alpha_1, \alpha_2 = 0##. I don't know...

We can make the first three functions add up to zero in the following way : ##\sin^2 t+\cos^2 t-\frac{1}{3}\times 3 = \varphi_1(t) + \varphi_2(t) - \frac{1}{3} \varphi_3(t) = 0##. However, look at ##\varphi_4(t) = t## and ##\varphi_5(t) = e^t##. How does one combine the two to add up to zero? I...

See attachment for identities and proofs, if you find my proofs are incorrect in some way please post it. Thanks for your time.

Why do we define functions as only as only those graphs which have only one y value for each x value. for eg. we don't say that a circle is a graph of a function,because its graph would have two y values for same x values. what i mean to ask is why not call anything that takes a input and gives...

Hi all, Let me give some background to my question. In computational neuroscience it's now fashionable to train recurrent neural networks (RNNs) to solve some task (working memory for example). We do so by defining some cost function and training the model weights to minimize it (using...

I am looking for the expectation of a fraction of Gauss hypergeometric functions. $$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$ Are there any identities that could be used to simplify or...

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 3.12...

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 3.12...

##\textbf{Attempt at solution}##: If I can show that ##f## is integrable on ##[a,b]##, then for the second part I get : Let ##\frac{\varepsilon}{b-a} > 0##. By definition of uniform convergence, there exists ##N = N(\varepsilon) > 0## such that for all ##x \in [a,b]## we have ##\vert f(x) -...

Other than for null-homotopic maps, which continuous maps defined on ##D^1 \rightarrow D^1## (Open disk)extend continuously to maps ##B^1 \rightarrow B^1 ## ,(##B^1## the closed disk) which maps can be extended in opposite direction, i.e., continuous maps ## f: S^1 \rightarrow S^1 ## that...

For historical reasons the hyperbola always was considered to be one of the «classical» curves. The function, obviously, does not belong to C0. Apparently, is does not fit L2 or any other Lp? What is the smallest class?

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 2: Differentiation ... and in particular I am focused on Section 2.1: Limits ... I need help with an aspect of the proof of Proposition 2.1.2 ...Proposition 2.1.2 and its proof read as follows: In the...

ok I got stuck real soon... .a find where the functions meet $$\ln x = 5-x$$ e both sides $$x=e^{5-x}$$ok how do you isolate x? W|A returned $x \approx 3.69344135896065...$ but not sure how they got itb.? c.?

Hey guys, I have written a C++ code which is based on two main classes: Particle and Group. Each Group contains a set of Particle(s), each Particle is defined by a set of coordinates, and has an associated energy and force (the energy/force evaluation is done by calling an external program). I...

I can only find a solution to \int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when m = n (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel...

How can we write (finite) nested expressions as compositions of functions? For example (using Horner's technique), consider: ##P(x) = 3 + 2x + 4x^2 + 6 x^3 = 3 + x(2 + x(4 + x(6) ) )## The way I see to do it is to use functions of two variables. ##f_3(x,y) = 6## ##f_2(x,y) = 4 + xy##...

Homework Statement:: Violin String Shape Functions Homework Equations:: Violin String Shape Functions Hello, Is anyone working on violin string shape functions(Timoshenko Beam Theory)? It would be really helpful to my research if we share our knowledge on this topic. Thank you

If I have a sum ##f(x) + g(x) = c##, with ##c## a constant, does this imply that both ##f(x)## and ##g(x)## are also constants? If I just solve this equation for ##x##, I will find some values of ##x## which satisfy the equation. However, if I require that the equation be true for all ##x##...

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 some further help in fully understanding the proof of Proposition 8.14 ...

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 some further help in fully understanding the proof of Proposition 8.13 ...

I can only find a solution to \int_{0}^{r} \rho J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . The closed form solution to \int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho I am not able to find anywhere. Is there any way in which I can approach this problem from scratch...

I already found ##I(t)## using Kirchhoff's laws, I got the equation ##V-RI-L\frac{dI}{dt}=0\Rightarrow L\frac{dI}{dt}=V-RI## then I solved the differential equation getting ##I(t)=\frac{V}{R}\left[1-e^{-\frac{R}{L}t}\right]##. My problem is founding the voltage as a function of time ##V(t)##, I...

Sorry if i made any language errors, english is not my first language. Question: An area in the first quadrant (x=>0,y=>0) is limited by the axis and the graphs to the functions f(x)=x^2-2 and g(x)=2+x^2/4. When the area rotates around the y-axis a solid is created. Calculate the volume of...

If we plot a list of functions in a literal array, they get plotted automatically in different colors, for example: Plot [{x,x*x,-x},{x,0,10}] But if we get the list of functions from another user-defined function, they get plotted in a single color: fnY[x_]:={x, x*x, -x} ... OR ...

Let f(x) = \sqrt{x} Assume that g is function such that (i) g(c)= c+m(x-1) (ii) f(1) = g(1), and (iii) \lim_{{x}\to{1}}\frac{f(x)-g(x)}{x-1} Answer the following questions. Show all of your work, and explain your reasoning. (a) What are the constants c and m? (b) How does g compare with the...

Hi PF! I'm trying to find the eigenvalues of this ODE $$y''(x) + \lambda y = 0 : u(0)=u(1)=0$$ by using the basis functions ##\phi_i = (1-x)x^i : i=1,2,3...n## and taking inner products to formulate the matrix equation $$A_{ij} = \int_0^1 \phi_i'' \phi_j \, dx\\ B_{ij} = \int_0^1...

In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...

Hi. I would like to check that my understanding is correct. For ##f(x)=x^{1/n}## where n is an integer. If n is odd then f(x) is an odd function while if n is even then f(x) is neither odd or even as it involves the square root function which is only defined for non-negative x. For ## f(x) =...

I have to transform the first function which is f(x)=x^3 to the second function. First, I have to find each shift then combine those to make a new function equation. I've used desmos and I know that there is a horizontal shift 3 units to the right. There is a vertical shift up but I don't know...

Hello, good morning. I would like to know if someone knows a program to be able to draw the functions in the same way as the one shown in the image and also allow me to point out an enclosure formed by them without having to use inequalities to do so. Thank you very much for everything beforehand.

I am reading the book: "Theory of Functions of a Complex Variable" by A. I. Markushevich (Part 1) ... I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows: At the start of the above proof by Markushevich we read the following: "If...

Hello, It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem: Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics? If so, what is the relationship of trigonometric functions...

The examples of "formal" power series and polynomials in one indeterminate are familiar and useful in algebra. However, I don't recall the example of rational functions (ratios of polynomials) in one indeterminate being used for anything. Is that concept useful? - or trivial? -or equivalent...

I have two different solutions, and I do not know which one is correct and why the other one is wrong. Solution 1. In the ##L_z## space, the spin state is ##\begin{pmatrix} \sqrt { \frac 4 5} \\ \sqrt { \frac 1 5} \end{pmatrix}##, and ##S_x=\frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0...

Hi, This is on the wikipedia entry for the Euler Lagrange equation. Here is a link. https://en.wikipedia.org/wiki/Calculus_of_variations#Euler%E2%80%93Lagrange_equation The notation I am confused about is this: Aren't the y(x) and the y'(x) unnecessary to list as arguments when x is...

Since, both the domain and the range is set of integers, we must have just operations of addition and multiplication only in the function. That means, function should be some kind of a polynomial. Plugging ##a=0## and ##b=0##, I can deduce that ##3 f(0) = f(f(0))##. Also I can deduce that ##3...

Hi, how would I go about drawing these two graphs? and The first one would be concentric circles with the centre at (0,0). The second one would be straight lines through (0,0). Is this correct? Also, what happens at ln(0) = constant for the first graph and x = 0 for the second graph...

I am reading Theodore W. Gamelin's book: "Complex Analysis" ... I am focused on Chapter 1: The Complex Plane and Elementary Functions ... I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ... The...

I'm trying to solve the vector potential of a solid rotating sphere with a constant charge density. I'm at a point where I'm performing the final integral that looks like $$ -\left( \frac {\mu_0 i} {3} \right) \sqrt{\frac 3 {2\pi}} \frac {q\omega}{R^3} Y_{1,1} \int_0^R (r')^3 \frac {r_<}...

As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations. However, what I noticed in Source #2 was that, when functions are represented as vectors, the...

Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...

I have a single technical question regarding a statement on page 7 of the paper "Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence". The paper up until page 7 defines a general correlation function ##\mathcal{G}## of a basic quantum Ising...

I fell upon such a wrting : $$du=tan(d\theta)$$ How to integrate this ? I didn't try numerically but I thought of expanding the tangeant in series but then should for example $$\int d\theta^2$$ be understood as a double integration ?

ok I have been trying to cut and paste in packages and code to get a simple inverse function to plot but nutin shows up and get error message. if possible I would like no grid but an xy axis with tick only where the graph goes thru the axis and of course a dashed line of x=y some of the...

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 $\quad \ln{x}+1-1=\ln{x}$...