Function Definition and 1000 Threads

Can anyone explain to me why the second one is the right? (See the attachment)

So, I've recently played around a little with the Gamma Function and eventually managed to find an expression for the Beta Function I have not yet seen. So I'm asking you guys, if you've ever seen this expression somewhere or if this is a new thing. Would be cool if it was, so here's the...

Suppose ##f## is holomorphic in an open neighborhood of the closed unit disk ##\overline{\mathbb{D}} = \{z\in \mathbb{C}\mid |z| \le 1\}##. Derive the integral representation $$f(z) = \frac{1}{2\pi i}\oint_{|w| = 1} \frac{\operatorname{Re}(f(w))}{w}\,\frac{w + z}{w - z}\, dw +...

I would like to understand the highlighted part. In my understanding, this function does not seem to have a hole! Having said this, i can state that ##x_0=1## and we have our defined ##f(x_0)=2##. It follows that, ##f(1^{+}) = e## ##f(1^{-}) = e## thus ##f(x_0^{+})=f(x_0^{-})≠f(x_0)## thus the...

Good morning, I need some help solving those two question. I've attached my attempted solution below. Could i solve the transfer function any further? Thank you for your help

Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos...

I am confused at how to code this without using any of matlab's already built in functions except for using double. Is this question just asking me to write out the function and then make sure it's double precision?

I'm trying to solve an improper integral, but I'm not familiar with this kind of integral. ##\int_{-\infty}^{\infty} (xa^3 e^{-x^2} + ab e^{-x^2}) dx## a and b are both constants. From what I found ##\int_{-\infty}^{\infty} d e^{-u^2} dx = \sqrt{\pi}##, where d is a constant and...

Earlier today, I posted a question about the strain energy function. I am happy with the answer (I love this group). But the answer opened up a deeper question. Many elasticity textbook posit the existence of a strain energy function: And they make an additional assumption about its...

This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.

I need to find the zeros of this function where d,L,v are constants. After several calculations I faced this equation. I tried everything I know, but I can't solve this. Maybe I'm missing something or I must made a mistake earlier in the problem. Thus, I would like to know if it is possible to...

This is the question: This is the ms solution- from Further Maths paper. My question is referenced to the highlighted part. I can see they substituted for the lower limit i.e ##x=1## to get: ##F(x)=\dfrac{x^3-1}{63}## supposing our limits were; ##2≤x≤4## would the same approach apply? Anything...

Greetings I have a hard time understanding how the radial tooth clutch function when it stops transferring power . Basically I understand that clutches: 1) transfer power from input shaft to output shaft 3) disengage when the torque transmitted has reached a certain limit ( normally when the...

I just came across this...the beginning steps are pretty easy to follow...i need help on the highlighted part as indicated below; From my own understanding, allow me to create my own question for insight purposes... let us have; ##7^x=5x+5## ##\dfrac{1}{5}=(x+1)7^{-x}##...

I have attached my attempt at a solution. In the solution image, I have computed 3 things: 1. System transfer function based on my understanding of the problem statement. This is a 2nd order system with steady state dc gain=0.9. So I wrote the transfer function accordingly. However, I strongly...

I know that TikZ is probably the best way to do this but most Forums don't use it. I can make a rectangular diagram, but it's a bit clunky: Say I have the commutative function diagram: ##\begin{array}{ccccc} ~ & ~ & f & ~ & ~ \\ ~ & A & \longrightarrow & B & ~ \\ g & \downarrow & ~ &...

I've attached what I have so far. Used Gauss's law, everything seemed to make sense except the units don't work out in the end. The charge density function if given by: r(z)=az, where z is the perpendicular distance inside the plane.

$$B(t) = B_{0} \frac{t^2}{T^2}$$ for ##0 \leq t \leq T## The issue here is more conceptual, because once I find the flux of B I know how to proceed to find the current. I got velocity (but it seems to me that it is the initial velocity), I could use it to find the time in function of space...

Greetings, is it possible to characterize a sinusoidal wave in the domain of time and then pass into the domain of movement along x direction? I start with: a is the amplitude of the sine function and ω is the angular velocity. t is the time. I can express the angular velocity in funct. of the...

Hi PF! When I execute the code below: import numpy as np from scipy.stats import t import scipy.optimize as optimizeglobal data data = np.random.normal(loc=50, scale=1, size=(2400, 1)).flatten() def L(F): M = 250 lmda = 0.97 sig_0 = F[0] for i in range(1, 12): sig_0 +=...

The known expression of the wave function is where A is the amplitude, k the wave number and ω the angular velocity. The mathematical definition of arc length for a generical function in an interval [a,b] is where, in our sinusoidal case: For our purpose (calculation of the length in one...

Hi PF I'm trying to minimize a function func via scypi's minimiz function, as shown below. import numpy as np import scipy.optimize as optimize def func(x): y = x[0]**2 + (x[1]-5)**2 print('hi') return y bnds = [(1, None), (-0.5, 4)] result = optimize.minimize(func, method='TNC'...

I've defined this function to clean up some pages of work I've been doing on relations of integers modulo n. Let's call it mav(a,n) for now. mav(a,n) for integers a and n is equal to the Euclidean distance from a to the nearest multiple of n. To compute it in programming languages I've been...

This is question for aerodynamicist, so I put it here in aerospace department. (Mechanical engineers don't learn aerodynamics at university) What is aerodynamic function of active front lip spoiler (on video) and what is function of flexibile plate infront of front tyers(picture)? Why reduce...

The textbook I am self studying says that the wave function for a free particle with a known momentum, on the x axis, can be given as Asin(kx) and that the particle has an equal probability of being at any point along the x axis. I understand the square of the wave function to be the probability...

Q. Calculate the linearised metric of a spherically symmetric body ##\epsilon M## at the origin. The energy momentum tensor is ##T_{ab} = \epsilon M \delta(\mathbf{r}) u_a u_b##. In the harmonic (de Donder) gauge ##\square \bar{h}_{ab} = -16\pi G \epsilon^{-1} T_{ab}## (proved in previous...

Dear Mr. and Ms., I am trying to measure the autocorrelation functions of 2D ising model based on the equation given by where A(t) denote a measure. I calculate a c(t) of magnetization. I calculated in this way data_path = f"../../trajectory/data.txt" data = np.loadtxt(data_path)...

Velocity as a function of time, defined with units attached (Quantity feature of Mathematica): fnVq[t_ ]:= 2 m/s^2 * t fnVq[5 s] Integrate[fnVq[tt],{tt,0 s, 2000 ms}] 10m/s 4m When we printed above the value and integral, we got the correct results with proper units. Now I'm trying to...

hi guys i found this problem in a set of lecture notes I have in complex analysis, is the following function real: $$ f(z)=\frac{1+z}{1-z}\;\;, z=x+iy $$ simple enough we get $$ f=\frac{1+x+iy}{1-x-iy}= $$ after multiplying by the complex conjugate of the denominator and simplification $$...

Hi PF! I created a function ##R(x)## that gives the gap between the largest two primes less than or equal to ##x##. To define it, I used this property: $$\pi(x+R(x))=\pi(x)+1$$ Which is true since the ##x## distance between ##\pi(x)## and ##\pi(x)+1## is ##R(x)##. If we solve for ##R(x)## we...

Let's say that we have a one-particle Hamiltonian that admits only a continuous spectrum of eigenvalues ##E(k)=\alpha k^2## parameterized by asymptotic momentum ##\mathbf{k}## (assuming the eigenfunctions become planewaves far from the origin), would the partition function then be $$Z=\int...

I'm trying to understand the function of the air cavity inside drums. I've read that 'The air cavity inside the drum will have a set of resonance frequencies determined by its shape and size. This will emphasize some frequencies at the expense of others.' Then what are the resonance...

The following is the wave equation from Electrodynamics: $$\frac{\partial^2 \Psi}{\partial t^2} = c^2\frac{\partial^2 \Psi}{\partial x^2}$$ Where ##\Psi## is the wave function. But because of Heisenberg's Uncertainty, physicists had to come up with another equation (the Schrodinger equation)...

This is the code that i wrote Clear["Global`*"] Z = 500; W = 100000; G = 250; H = 100; K = 0.5; T = 30; L = 4000; P = 5; S = 2.5; Y = 1; A = 0.1; V = 2.5; J = 8000; f[x_] := 1/ x {(J*Z*x*(2*Y - x))/( 2*Y) - ((W + T*G) + ((L + T*P)*2*Z*Y*(1 - ((Y - x)/Y)^1.5))/ 3 + (H + T*S +...

The Korean textbook standard defines the convexity of the function as an open section. Many textbooks and university calculus textbooks define the convexity of the curve as an open section. However, some textbooks define convexity as closed sections. Do you think it is right to define the...

For a nonconservative force, What would be the dissipative function for a force f=-bvⁿ in Lagrangian (Where v is the velocity) [#qoute for a nonconservative force f=-bv The dissipative function is D=-(1/2)bv² ] Since ##f=\frac{\partial D}{\partial \dot x}## so the dissipative function should...

During a thermodynamic cycle, an ideal thermal machine absorbs heat Q2 > 0 from a hot source and uses it to perform Work W > 0, giving a cold source a heat Q1 < 0 with an efficiency of 20% . How much is the work done as a function of Q1 ?I have 2 question regarding this problem: 1) Why is Q1 the...

I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x): where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...

Hi, I am trying to get the transfer function from a wall between rooms. From one side I have the force of a hammer as an input ,and in the other side of the wall (next room) I have an accelerometer. Is it possible to get the TF without know the damping, stiffness and mass of the wall partition...

program main ! use ! some library that defines the function to calculate the determinant of a given matrix implicit none real,dimension(2,2)::A real::det_val A(1,1)=1 A(2,2)=1 A(2,1)=0 A(1,2)=0 ! det_val=det(A) print *,det_val ! Should print 1. end program main

How do spinors affect wave function solutions? Like how is the output different

Hello! Consider this filter,and that I have to find the transfer function U2/U1 with the norm Ω= ωRC (also double fractions are not allowed) Now I can see that that the resistor and capacitor left as well as right are parallel to each other. So simplifying that ## Z_1 = \frac{R}{jωRC +1}...

Now, I don't understand how did author compute $F_{X_1}(x) = \displaystyle\sum_{j=1}^n \binom{n}{1} F^1(x) (1-F(x))^{n-1} = 1-(1-F(x))^n ?$ (I know L.H.S = R.H.S) Would any member of Math help board explain me that? Any math help will be accepted.

Author computed ##f_{X_k}(x)## as follows but I don't understand it. Would any member explain me the following computations?

Hi everyone This is the solution for the problem. I don't understand how they got from To This was my attempt at a solution I can't seem to get rid of one of the y terms and am left with one on each side. Could someone explain the solution to me please? Thanks

I have a table of values (from my own analysis, not from a textbook) that represents a portion of a periodic function: x y 0 30 1 20 2 10 3 10 4 20 5 30 What function satisfies the table? What I know is that the function is periodic. I was thinking I could use cosine because its...

In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away) Starting from a five-dimensional embedded hyperboloid (as in carroll pg. 324) ## -u^2 + x^2 + y^2 + z^2 + w^2...

Hello everyone! I was doing some dimensional analysis to find an equation that gives a acceleration as a function of time, using constant power. I came up with the equation $$a = k\sqrt {\frac P {mt}}$$ I differentiated velocity with respect to time in order to check my work and also checked out...

In hamiltonian formalism we have the generalized coordinates ##q_i## and the conjugates moments ##p_i##. For a dipole in a give magnetic field ##B## the Hamiltonian is ##H=-\mu B cos \theta## where ##\theta## is the angle between ##\vec \mu## and ##\vec B##. Can i consider ##\theta## or ##cos...

Hello! I have this filter here a) Calculate the transfer function T(Ω) = Ua/Ue using voltage dividers.For this, use the normalized angular frequency Ω = ωRC and bring the result into the form ##T(Ω) = \frac{A+jB}{C+jD} ## . The result must not contain any double fractions. I was able to that...