Function Definition and 1000 Threads

I think the answer is an even function as the function ##x^2## is an even function and thus, is symmetrical w.r.t. Y axis. The question I have is how to do this problem algebraically. I tried to graph some functions on GeoGebra to verify my answer. a) ##y = ln(x^2)## b) ##y = sin(x^2)##...

There is a typo. It should say ##h=\frac{f}{g}##. Attempt: ##f## and ##g## are holomorphic on ##\Omega##. Homomorphic functions form a ##\mathcal{C}^*## algebra, so ##h## is holomorphic on ##\Omega## where ##g\neq 0##. If ##z_0## is a removal singularity of ##h##, then ##Res(h,z_0)=0## by...

Hello to all. I am trying to design a ripple free (read as ripple free as possible) power supply (PS) for my DIY DDS function generator. I am (was) in the possession of the hyland 5v to 12v PS which wrecked due to a stupid action on my side, my bad. so i was going to repair it, but i found that...

First of all I am not sure which type of singularity is ##z=0##? \ln\frac{\sqrt{z^2+1}}{z}=\ln (1+\frac{1}{z^2})^{\frac{1}{2}}=\frac{1}{2}\ln (1+\frac{1}{z^2})=\frac{1}{2}\sum^{\infty}_{n=0}(-1)^{n}\frac{(\frac{1}{z^2})^{n+1}}{n+1} It looks like that ##Res[f(z),z=0]=0##

Why do we want to always deal with single valued functions? In the classical treatment a function is a rule which assigned to one number another number. In the modern sense, it is a rule which assigns to each element in a set called the domain an element (one element) in a set called the range...

I don't understand why my solution is wrong. Here is my solution. $$ r_{\theta} = R\cos{\theta} \vec{i} + R\sin{\theta} \vec{j} $$ $$ v_{\theta} = v\cos(\theta + \frac{\pi}{2}) \vec{i} + v\sin(\theta + \frac{\pi}{2}) \vec{j} $$ $$ p_{\theta} = mvR(-\sin{\theta}) \vec{i} +mvR(\cos{\theta}...

False The reasoning for answer: The absolute value function is is not analytic wherever its argument equals zero. ##f## is not analytic at ##z=0## so it is not entire.

I've never actually done this, so I was wondering if someone could show me how this is done. One way I tried was by simply using ##cos^{-1}## in order to cancel the cosine, but that gave me a different value, so I assume this is not how you are supposed to do this. --> I know I am supposed to...

I need to know how to predict particle size of a water driplet produced by a given ultrasonic frequency? For example, an ultrasonic fogger will create ~5 micron water driplets at a frequency of 1.75 MHz. I do know that the higher the frequency the smaller the driplet diameter. How is this...

Here is what the problem looks like. The thing is I don't remember what π1is exactly and I don't really know much group theory or know what equivalence classes are. I remember learning some group theory fact that f*(n) = n*f*(1). So, I think (a) was just equal to m since f(1) = 1 and (b) was...

Has any path/line/shape/contour function? And how find function of complex "path"? for example this?

Hi, I have a class master_t which is composed by two other classes, dev_a, dev_b. I would like that a member function from the dev_b object (within master_t) could use a member function of dev_a object (within master_t). This is a minimal working code, where line 26 implements this feature...

I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...

I'm stuck in a part of my problem where I need to find the roots of this function which represent turning points for a non-ideal spring.

Consider item ##vii##, which specifies the function ##f(x)=\sqrt{|x|}## with ##a=0## Case 1: ##\forall \epsilon: 0<\epsilon<1## $$\implies \epsilon^2<\epsilon<1$$ $$|x|<\epsilon^2\implies \sqrt{|x|}<\epsilon$$ Case 2: ##\forall \epsilon: 1\leq \epsilon < \infty## $$\epsilon\leq\epsilon^2...

For a Prandtl stress function to be valid, it must be zero on the boundary. For a circular bar, both of these work: $$\phi_1 = C\left(\frac{x^2}{r^2}+ \frac{y^2}{r^2} - 1\right)$$ $$\phi_2 = C \left(x^2+ y^2- r^2\right)$$ But performing the integration for the internal torque M gives...

(a) i tried to decompose the fracion as a sum of fractions of form ##\frac{1}{1-g}## $$f=\frac{-z}{(1+z)(2-z)}=\frac{a}{1+z}+\frac{b}{2-z}$$ $$a=\frac{1}{3}, b=-\frac{2}{3}$$ $$f=\frac{1}{6}\frac{1}{1+z}-\frac{1}{3}\frac{1}{1-\frac{z}{2}}$$ $$f=\frac{1}{6}\sum_{n=0}^\infty...

I learned that ##f## has another singular point at ##z=1.715##, but i don't think this would be related to the pole at ##z=0## I tried substitutine ##u=2\cos z-2+z^2## and $$f(u)=\frac{1}{u^2}$$ has a pole of order 2 at ##u=0## which happens i.f.f. ##z=0## or ##z=1.715##. so ##f## has a pole...

Dear Everybody, I need some help understanding how to use pade approximations with a given data points (See the attachment for the data). Here is the basic derivation of pade approximation read the Derivation of Pade Approximate. I am confused on how to find a f(x) to the data or is there a...

I am totally confused about the task. Any help will be nice. Thank you so much

Hi all In the Lagrangian, we have L = KE - PE In most cases, I have seen KE as a function of q and q-dot (using the generic symbols). However I first learned how KE = 0.5 m * v-squared. Later, I used generalized coordinates and THAT is when KE became a function of q. I get all that...

The following solves an IVP, giving the output as the function f3[x]: s3 = NDSolve[{(-z1[t]^(3/2) + (1 + z1[t]^2)^(3/4))/( 3 (-z1[t] + Sqrt[1 + z1[t]^2])) == z1[t] z1'[t], z1[0] == 0.0001}, z1, {t, 0, 30} f3[x_] := z1[x] /. First[s3]; My question is, how do I curve fit f3[x] to the...

Part 1 ##\left\| \vec{y} \right\|^2 \leq \left\| \vec{y} \right\|^2## and since ##\lambda \in \left[ 0,1 \right] \Rightarrow \lambda^2 \leq \lambda## So ##\lambda^2 \left\| \vec{y} \right\|^2 \leq \lambda \left\| \vec{y} \right\|^2 ## Part 2 ##\left\| \vec{x} \right\|^2 \leq \left\| \vec{x}...

Hello, I'm struggling with this for some time. So I have the function: f(x) = sqrt(1 - 1/x) The derivative of this function can be easily calculated. Now we define the function: F(x) = f(x)/f(x + dx) = sqrt(1 - 1/x)/sqrt(1 - 1/(x+dx)) I have a hard time to find F'(x) due to the presence of...

Why does f attain its local maximum at r' in (p,q). Is it because we have f(x)<= f(r') for all x in (p,p+delta)?

Consider two different Taylor expansions. First, let ##f_1(s)=(1+s)^{1/2}## $$f_1'(s)=-\frac{1}{2(1+s^{3/2})}$$ Near ##s=0##, we have the first order Taylor expansion $$f_1(s) \approx 1 - \frac{s}{2}$$ Now consider a different choice for ##f(s)## $$f_2(s)=(1+s^2)^{1/2}$$...

[FONT=times new roman]Problem : [FONT=times new roman]We are required to show that ##I = \int_C x^2y\;ds = \frac{1}{3}##. Attempt : Before I begin, let me post an image of the problem situation, on the right. I would like to do this problem in three ways, starting with the simplest way - using...

Hi, PF Although a function cannot have extreme values anywhere other than at endpoints, critical points, and singular points, it need not have extreme values at such points. There is an example of how a function need not have extreme values at a critical point or a singular point in 9th edition...

From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...

The final wave function solutions for a particle trapped in an infinite square well is written as: $$\Psi(x,t) = \Sigma_{n=1}^{\infty} C_n\sqrt{\frac{2}{L_x}}sin(\frac{n\pi}{L_x}x)e^{-\frac{in^2{\pi}^2\hbar t}{2m{L_x}^2}}$$ The square of the coefficient ##C_n## i.e. ##{|C_n|}^2## is...

Regarding the electrical permittivity of the metal in a high frequency regime, I cannot find research material related to the lead dielectric function (PD). I can't get the matatrial as values, I'll let you comment on that. I know that Pd can inhibit the amount of gamma rays in the x-ray case...

Hi all, I am currently studying renormalization group and beta functions. Since I'm not in school there is no one to fix my mis-understandings if any, so I'd really appreciate some feedback. PART I: I wrote this short summary of what I understand of the beta function: Is this reasoning...

Let's say we have a function ##M(f(x))## where ##M: \mathbb{R}^2 \to \mathbb{R}^2## is a multivariable linear function, and ##f: \mathbb{R} \to \mathbb{R}^2## is a single variable function. Now I'm getting confused with evaluating the following second derivative of the function: $$ [M(f(x))]''...

hi guys I came across that theorem that could be used to check if a surface represented by the function f(x,y,z) = λ could represent an equipotential surface or not, and it states that if this condition holds: $$\frac{\nabla^{2}\;f}{|\vec{\nabla\;f}|^{2}} = \phi(\lambda)$$ then f(x,y,z) could...

A rocket of initial mass m0 is launched vertically upwards from the rest. The rocket burns fuel at the constant rate m', in such a way, that, after t seconds, the mass of the rocket is m0-m't. With a constant buoyancy T, the acceleration becomes equal to a=T/(m0-m't) -g. The atmospheric...

Define a continuous function $$F(x;n)$$ that interpolates points (x, x mod n) for a given integer n and all integer x. For example $$F(x;2)=\frac{1}{2}-\frac{1}{2}\cos\left(\pi x\right)$$ interpolates all points (x, x mod 2) when x is an integer. Similarly $$F(x;3)$$ should interpolate points...

I have a differential equation of the form y''(t)+y'(t)+y(t)+C = 0. I think this implies that there are non-zero initial conditions. Is it possible to write a transfer function for this system? This post...

To find the tension in the rope connecting 6.0 kg block and 4.0 kg block we do 6.0 kg = m1, 4.0 kg = m2, 9.0 kg = M (m_2 + m_1)a - Ma = Mg - m_2 gsin\theta - m_1 gsin\theta Why do we use sin in these equations and not cos?

let ##X=\{0,p1,p_2,...,p_n,1\}## and ##Y=\{0,p1,p_2,...,p_n,1\}## be sets equipped with the discrete topology. for each ##q_i## in ##Y##, the inverse image ##h^{-1}(q_i)=p_i## is open in ##X## w.r.t. to the discrete topology, so h is continuous. every element y in Y has a preimage x in X, so h...

Let f be continuous in [0,1] and g be continuous in [1,2] and f(1)=g(1). prove that $$ (f*g)= \begin{cases} f(t), 0\leq t\leq 1\\ g(t), 1\leq t \leq2 \end{cases}$$ is continuous using the universal property of quotient spaces. Let ##f:[0,1]→X## and ##g:[1,2]→Y## f and y are continuous, thus...

I included this image because it is easier than typing it out. Anyway, this is an old problem I need to catch up on. I have a clue as to how to do part a. I could say given an x that is a member of ∩V(Ai) which implies that x is a member of V(Ai) for ∀i. Then we can say ∀i all polynomials are in...

Hi, I have a question regarding the envelope function in parameter representation. Let an array of curves in cartesian coordinates be given in parameter representation, with curve parameter 𝑡 and array variable 𝑐 𝑥=𝑥(𝑡,𝑐) 𝑦=𝑦(𝑡,𝑐) Condition for envelope is: 𝜕/𝜕𝑡 𝑥(𝑡,𝑐) 𝜕/𝜕𝑐 𝑦(𝑡,𝑐)=𝜕/𝜕𝑐...

Im confused about a certain part of solving an equation. So I used the hyerbola formula to find the answer but I think I did the math wrong. X^2-y^2=c^2 X=1 Y= (2x^5-1)^2 I did the calculations as you can see in the picture but I know I messed up on the square root part. When you square one...

Summary:: We are currently studying basics of quantum mechanics. I'm getting the theory part but it's hard to visualise everything and understand. We are given this question to plot the function so if someone could help me in this. Plot the following function and the corresponding g²(x) g(x)...

Let a spherical wave propagate from the origin, $y = ADcos(wt-2\pi r/ \lambda)/r$. Also, let a plane wave propagate parallel to the x axis, $y = Acos(wt-2\pi r/ \lambda)$. At x = D there is a flat screen perpendicular to the x axis. Find the interference at the point y on the screen as function...

I'm curious how close someone could get to guessing the functions that generated the data shown below. And also, without looking at the plot, what do you think would be the most interesting looking function of x,y,z you can think of. A) B) C)

Basically with this problem, I need to show that f is continuous if A and B are open and if A and B are closed. My initial thoughts are that in the first case X must be open since unions of open sets are open. My question is that am I allowed to assume open sets exist in Y? Because then I can...

Hello, To first clarify what I want to know : I read the answer proposed from the solution manual and I understand it. What I want to understand is how they came up with the solution, and if there is a way to get better at this. I have to show that, given a vector field ##F## such that ## F ...