Coefficients Definition and 753 Threads

I'm having trouble finding a definite answer to this question: When finding the Fourier series of a function is it always possible to find ##a_0## by first finding ##a_n## and just plugging in ##n=0##?

Evaluate sum: $\displaystyle S=\sum_{k=0}^{2n}(-1)^k{2n\choose k}{4n\choose 2k}$

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- \frac{4}{3} \zeta(3) $$ where $\psi_{1}(x)$ is the trigamma function. But I can't get my answer in...

I am looking for a counting interpretation to make the following identity evident: \sum_{k=0}^{n-j}(-1)^k\binom{j-1+k}{j-1}\binom{n}{j+k} = 1 The form of it looks like inclusion-exclusion. The sum is 1, more or less independent of j. So that makes me think it would be something like "how...

Hello everybody, I want to determine the extinction coefficients of Fe2+ and Fe3+ in glass. There are literature data (e.g. Weyl's book "coloured glass"), so I know what kind of curves I should expect. As I am studying a slightly different soda-lime-silicate system, I want to recalculate the...

Homework Statement Let \hat{u}_k the Fourier coefficients of 2-periodic function u(t)=t with t\in [0,2). Evaluate the sum of the serie: \sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2 Ok, I think there is a trick that I don't know... \sum_{k=-\infty}^{\infty}\hat{u}_k...

Homework Statement Metal sphere of radius R1 is surrounded by a concentric metal shell of inner radius R2 and outer radius R3. The dielectric is air. Calculate coefficients of capacitance for the given setup. Homework Equations Picture of the problem: The Attempt at a Solution...

When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for k_1and k_2 both zeroes of J_n(t), the orthogonality relation given by: $$\int_0^1 J_n(k_1r)J_n(k_2r)rdr = 0, (k_1≠k_2)$$ and for k_1 = k_2 = k: $$\int_0^1 J_n^2(kr)rdr = \frac12J_n^{'2}(k)$$ I...

y(4) - y = 3t + cos t y''' - y' = 2sint y(4) + 2y'' + y = 3 + cos 2t For problems like these, how do you determine what to make Y(t) for the trigonometric part? Sometimes its At cos t, A cos t, A sin t, and I think occassionally even A0sin t + Acos t Am I simply supposed to guess until...

Help slove pls!:( A uniform smooth sphere P,of mass 3m, is moving in a straight line with speed u on a smooth horizontal table.Another uniform smooth sphere Q, of mass and m and having the same radius as P, moving with speed 2u in the same straight line as P but in the opposite direction to P...

I created a little model that predicts the time a task will take to execute. Than, I calculate the real time that the task took to execute. With the predict and real value I have the error of my prediction. I want to use that error to predict the next execution (the next execution is a similar...

If one has a simple variable coefficient process like y'(t) = r(t)*y(t), is there a way to control it to a set point by feedback hitting the variable coefficients in r(t)? I am interested in feedback control of population processes. y'(t) = r*y(t) is simple proportional growth with a...

Hi all, In thermodynamics one often has equations like A dx + B dy = ∂f/∂x dx + ∂f/∂y dy From which follows A = ∂f/∂x B = ∂f/∂y Can anyone explain to me why this conclusion is necessary from a mathematical point of view, please? Here is my try: A dx + B dy = ∂f/∂x dx + ∂f/∂y...

If we have a metric connection with no other restrictions (torsion may be non-zero), do the connection coefficients {\Gamma^{\alpha}}_{\mu\nu} have any symmetries among the indeces? I'm thinking not. Or.. for a Levi-Civita connection, the only fixed symmetry condition is...

Homework Statement Find the Fourier coefficients for the function *Should be a piecewise function, not sure how to write one in [itex /itex] tags* f(x) = |x|, |x| < 1, 1, 1≤|x|< 2; f(x+4) = f(x) and Find the Fourier series for f(x) = cos1/2\pi x, -1≤x<1...

The annihilator method can be used to derived the entries in the following table for the method of undetermined coefficients, familiar to all students of ordinary differential equations: Type $g(x)$ $y_p(x)$ (I) $p_n(x)=a_nx^n+\cdots+a_1x+a_0$ $x^sP_n(x)=x^s\left(A_nx^n+\cdots+A_1x+A_0...

Not sure if this is the right place to post (but its related to a complex analysis questions) I'm doing a past paper for my revision and am stuck at the first hurdle. I simply cannot factor this polynomial in z for the life of me. I've tried completing the square and the usual quadratic...

Homework Statement Have a solution for the temperature u(x,t) of a heated rod, now using the orthogonality relation below show that the coefficients a_n , n = 0,1,2,... can be expressed as: a_n = \frac{2}{L} \int_{0}^{L} cos\frac{n\pi x}{L} f(x) dx Homework Equations \int_{0}^{L}...

I'm trying to find a comprehensive list of the empirical coefficients to be used in the following equation for calculating ideal gas constant pressure heat capacities: \frac{c^{IG}_P}{R}=A+BT+CT^{2}+DT^{-2}+ET^{3}(Eqn. 1) cPIG is the ideal gas constant pressure specific heat capacity; R is...

Hi, I'd just like to have a quick clarification with regards to the method of undetermined coefficients. I know that if a characteristic equation has the form (r-4)3 = 0 then the characteristic solution will be yc = e4t + te4t + t2e4t + t3e4t and the particular solution ought to be...

I'm doing a problem where I need to know the coefficients to change from the \vec{J} = \vec{J}_1 + \vec{J}_2 + \vec{J}_3 to the {\vec{J}_1, \vec{J}_2, \vec{J}_3} for three spin-1 particles, but I'm having trouble finding a table or reference for this... surely every time someone needs to...

Homework Statement Find the general solution for the following differential equation: y'' + x(y')^2 = 0. Homework Equations Integration, differentiation... The Attempt at a Solution Usually for these sort of DE you could use the substitution v(x) = y'(x) and this would simplify...

Homework Statement y''+y=3sin2t+2tcos2t Homework Equations The Attempt at a Solution I tried doing this problem by breaking it into 3 others. It doesn't quite work, and I want to know why. First, I solved for the nonhomog part of: y''+y=3sin2t Which is Y1 = -sin2t...

Here is the question: Here is a link to the question: Let P(x) = x4 + ax3 + bx2 + cx + d. Need to find the values of the variable that satisfy the guidelines below? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement A pullback toy car is pulled back. How far will it go before it stops? x(0)=0 v(0)=0 a(x<x0)=-k(x0-x) a(x>x0)=μg v(tf)=0 tf=D Homework Equations Fs=-kx F=mg Ff=μF The Attempt at a Solution d2x/dt2=1/m(k(x0-x)-μmg) Later in the run spring constant...

Hi when using the WKB approx, is there a general method to find these Refelction and Transmission coefficients, I have tried looking in books and on the net and I can't find a 'general' formula, they tend tjust to be quoted. I know that |T|^{2}+|R|^{2}=1 . And generally that T=...

Hi everyone. Is there any way to set demands on least square solutions? I have an equation on the form Ax=b, which is solved for x as: x=(A'*A)^-1*A'*b I do know for a fact that all values in x should be positive, but the least square solution for my particular system contains a...

There isn't a specific problem that's making me stuck, but I was hoping if someone could point me in the right direction here. I've looked up the topic online, but most of what I could find was through another approach or very unclear. The book I'm using also does not use the method my professor...

Homework Statement (2x-y)dx+(4x+y-6)dy=0 I need to solve by recognizing that I have linear coefficientsHomework Equations 2x-y=0 4x+y-6=0 Solving this system of equations I get x=1 and y=2The Attempt at a Solution After finding my intersection point to be (1,2) I let u=x-1, v=y-2. Which gives...

Homework Statement y''+2y'+y=xe-x Homework Equations Yc=c1e-x+c2xe-x relevant info on textbook: "If any term of yp is a solution of the complementary equation, multiply yp by x (or by x2 if necessary)." >> i don't understand the part where it says "a solution of the complementary equation"...

Hello everyone, I'm reading a bit about Clebsch-Gordan coefficients and I found two things in their general description I didn't quite understand. Can anyone help me with these questions? First, I read that the Clebsch-Gordan coefficients are zero unless the total angular momentum...

I am not sure how to do this one. Nothing I try goes anywhere. Consider the two-body equation of motion in vector form $$ \ddot{\mathbf{r}} = -\mu\frac{\mathbf{r}}{r^3}. $$ Show that the $f$ and $g$ functions defined by $$ \mathbf{r} = f\mathbf{r}_0 + g\mathbf{v}_0 $$ satisfy $$ \ddot{f} =...

Hi all, How do I compute the Fourier series coefficients for unit periods for cos(pi)x, the interval is from -1/2 to 1/2. I know the formula but I am getting a wrong answer ?

Homework Statement I'm supposed to show that the first, second and third differential coefficients of the field midway is zero. The B-field at that location is given by: which is field due to a ring x 2.. The Attempt at a Solution I tried to differentiate it with respect to x...

You testify as an expert witness in a case involving an accident in which car A slid into the rear of car B, which was stopped at a red light along a road headed down a hill. You find that the slope of the hill is θ = 12.0°, that the cars were separated by distance d = 23.0 m when the driver of...

Homework Statement Third question of the day because this assignment is driving me crazy: Suppose that \left\{ f_{k} \right\} ^{k=1}_{\infty} is a sequence of Riemann integrable functions on the interval [0, 1] such that \int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow...

So, until now I know: (DV/DS)p=(DT/Dp)s=a*T/cp*(rho) (enthalpy) (Dp/DT)v=(DS/DV)t=-a/k (helmoltz) (DS/Dp)t=-(DV/DT)p=-Va (gibbs) a=expansion coefficient k=isothermal compression coefficent cp=heat capacity at constante pressure I want to deduce DT/DV at constant entropy=(DT/DV)s. BUT...

Hi, The Fourier series can (among others) expressed in terms of sines and cosines with coefficients a_n and b_n and solely by sines using amplitudes A_n and phase \phi_n. I want to express the latter using a_n and b_n. Using a_n = A_n \sin(\phi_n) \\ b_n = A_n \cos(\phi_n) I...

Homework Statement Consider 2 conductor spherical shells of radii a and b (where a>b). The inner shell is at zero potential and the outer shell is at a potential given by ##V(\theta, \phi )=V_0 \sin \theta \cos \phi ## where ##V_0## is constant and theta and phi are the usual spherical...

Hi everybody! I am working in a solid state physics group and need to find some heat transfer data to finish my calculations. I need to find the Nusselt number Nu=\frac{hL}{K} describing the heat transfer interface between a Si wafer and helium gas at a variety of temperatures. To find this I...

Why we always write equation in form y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write: m(x)y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write coefficient ##m(x)## for example?

Homework Statement I've reached a relation but then I need to obtain the coefficients ##A_m## and ##B_m##'s, those are the only unknowns. Here's the expression: ##\sum _{m=0}^\infty a^m [A_m \cos (m \theta ) + B_m \sin (m \theta )]=T_0\sin ^3 \theta##. Homework Equations Fourier...

Homework Statement Hello guys. I'm totally stuck at finding the solution to ##y'''-12y'+16y=32x-8##. Homework Equations Variation of parameters once I'm done with the general solution to the homogeneous ODE. The Attempt at a Solution First I want to solve the homogeneous ODE...

For the wave equation I managed to get the coefficient of f: a1=2 and the coefficient of g: \frac{12pi}{2pi*2}=B2 Is these answers right, since my B2 does not match the answer I was given. Thank you

I have to understand some papers concerning pzt actuated membranes for a project and I keep stumbling upon d31 or d33 strain coefficients. Can someone please explain to me what do the 31 and 33 numbers refer to, regarding the geometry? I'm not sure I'm getting this so please feel free to ask...

Homework Statement I must find the oscillations of a circular membrane (drum-like). 1)With the boundary condition that the membrane is fixed at r=a. 2)That the membrane is free. Homework Equations The wave equation \frac{\partial ^2 u }{\partial t^2 } - c^2 \triangle u =0...

Given the ODE of the form: y''(x) + A*y'(x) + B*y(x) = 0 If we choose a solution such that y(x) = e^{mx} and plug it into the original ODE, the ODE becomes: (m^{2} + A*m + B)e^{mx} = 0 If we solve for the roots of the characteristic equation such that m = r_{1}, r_{2} (root 1 and root 2...

$$ A\exp\left(\frac{-t}{2}\right)\sin x\cosh \left(\frac{\sqrt{5}}{2}t\right) + B\exp\left(\frac{-t}{2}\right)\sin x\sinh \left(\frac{\sqrt{5}}{2}t\right) $$ What would be the guessed form using the method of undetermined coefficients? Also, if I had $$...

Using the coefficients determined, combine the terms to arrive at the following approximation for the solution $$ u(r,\theta,\phi)\simeq 25 + \frac{75\sqrt{2}}{2}r\sin\theta\cos\phi + \frac{125}{\pi}r^2\sin^2\theta\cos 2\phi. $$ How do I combine them? The coefficients are For $\ell = 0$ and $m...

Homework Statement Given the function 10sin^2(10t) Find the fundamental frequency and period. Find the exponential and trigonometric coefficients of the Fourier Series. Homework Equations The Attempt at a Solution I really have no idea how to start this problem. The sin^2...