Coefficients Definition and 753 Threads

Proof: Let ## f(x) ## be a function of the real variable ## x ## such that the integral ## \int_{-\pi}^{\pi}f(x)dx ## exists and if the Fourier coefficients ## (a_{n}, b_{n}) ## are defined by ## a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx dx, n=0, 1, ..., ## and ##...

Vide supra is a first few stages of Pascal's Triangle. The numbers themselves are the coefficients for the binomial expansion of ##(a + b)^n##, where ##n \in \{0, 1, 2, ... \}##. I'm just curious whether each stage has a special meaning, unique to it. My lack of knowledge prevents me from giving...

Margules suggested a power series formula for expressing the activity composition variation of a binary system. lnγ1=α1x2+(1/2)α2x2^2+(1/3)α3x2^3+... lnγ2=β1x1+(1/2)β2x1^2+(1/3)β3x1^3+... Applying the Gibbs-Duhem equation with ignoring coefficients αi's and βi's higher than i=3, we can obtain...

The output of SPSS 27 Canonical Correlation gives the standardized and unstandardized canonical correlation coefficients. What exactly are the standardized and unstandardized canonical correlation coefficients and what is the difference between them?

Hello, this is a very specific question so any help is much appreciated! GOAL: I'm trying to get a first-pass analytical approximation for the lift and drag coefficients for hypersonic flow over a blunt-body capsule spacecraft (similar to NASA's Apollo or SpaceX's Dragon) during atmospheric...

Hello, In studying linear regression more deeply, I learned that scaling play an important role in multiple ways: a) the range of the independent variables ##X## affects the values of the regression coefficients. For example, a predictor variable ##X## with a large range typically get assigned...

The implementations for the two filters in simulink are as follow: For the first filter: For the second one: The obtained results have values of 10^-12, while the expected results should be between 10^-3 - 10. Since it's the first time when I try t implement a tf with variable coefficients I...

I would like to arrive at the following expression for the quantity ##o_{\ell}## ( with "DM" for Dark Matter ): ##o_{\ell}=b_{s p}^2 C_{\ell}^{D M}+B_{s p}## with Poisson noise ##B_{s p}=\frac{1}{\bar{n}}(\bar{n}## being the average number of galaxies observed). the index "sp" is for spectro...

I have plotted the function for ##T=15## and ##\tau=T/30## below with the following code in Python: import numpy as np import matplotlib.pyplot as plt def p(t,T,tau): n=np.floor(t/T) t=t-n*T if t<(2*np.pi*tau): p=np.sin(t/tau) else: p=0 return p...

I have found that μstatic= 5.6 (found on graph) / 18.20 = 0.3077 I don't know how to find Force of friction from the graph to calculate μKinetic

I would try to determine whether ##p(t)## is even or odd. This would be so much easier if the values of ##\tau## and ##T## would be specified, but maybe it's possible to do without it, which I'd prefer. If for example ##\tau=1/2## and ##T=2\pi##, then ##p(t)=\sin{(2t)}## for ##0\leq t <\pi ##...

Hi all, I have another second order ODE that I need help with simplifying/solving: ##p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0## where ##\gamma,A,F## can all be assumed to be nonzero real numbers and ##D## is a purely nonzero imaginary number. Any help would be appreciated!

Do I determine this from the friction coefficients? Such that because mu_b > mu_a I know that it'll push on B because it has a bigger friction coefficient. such that this has nothing to do with the mass of the blocks? or does it? f.e. can I take it to be in an arbitrary direction and then...

Sometimes in calculations authors uses \frac{1}{h_1h_2}=\frac{h_3}{h_1h_2h_3} where ##h_i, i=1,2,3## are Lame coefficients. For instance in spherical coordinates ##h_r=1##, ##h_{\theta}=r##, ##h_{\varphi}=r\sin \theta##. I am not sure how we can divide so easily Lame coefficients when some on...

I don't understand why the highlighted term is there. This image was taken from Sean Carroll's notes available here: preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf

-1 -2 3 | 1 2 4 9 -31 15 | -1 -7 5 1 2 -3 7 9 -31 15 7 14 -21 -5 -10 15 5 -10 15The quotient has the coefficients -1, -7, 5. Now here is the problem. I assumed the variables would thus be...

I am trying to find the specific heat (at constant pressure) ##C_p## coefficients linked to the JANAF model, which basically assumes that ##C_p## is a polynomic function of ##T##, for liquid nitrogen (at ##\approx## 97 K). Before doing that, I am trying to find those for water (at ##\approx##...

I have seen two expansions of a vector potential, $$\mathbf A=\sum_\sigma \int \frac{d^3k}{(16 \pi^3 |\mathbf k|)^{1/2}} [\epsilon_\sigma(\mathbf k) \alpha_\sigma (\mathbf k) e^{i \mathbf k \cdot \mathbf x}+c.c.],$$ and $$\mathbf A=\sum_\sigma \int \frac{d^3k}{ (2 \pi)^3(2 |\mathbf k|)^{1/2}}...

I am asked to find Ro, a, and b. Th problem states the values are determined by the measurements at the normal ice, steam and sulfur points. So I approached the problem by plugging the the temperature problems. For 0°C, Ro reduces to 7 ohms. Then for the other two non zero temperatures, it looks...

What would happen with Einstein's coefficients if we wouldn't neglect the recoil effects?

I have solved a differential equation whose solutions is $$u = B + \sum_{n=1} C_{n }e^{-\lambda_{n}² q² t} J_{0}(\lambda_{n}r)$$ Where ##(\lambda_{n}r)## is such that ##J_{0}'(\lambda_{n}a) = 0##. So i should now try to satisfy the condition that, at t=0, u = ##f(r)##. The problem is that i...

I would like to attached a picture here...

Hello everyone, I am struggling with calculating the coefficients for second order transient analysis. For example, when analyzing a underdamped circuit, we know that the equation for voltage or current is xt=e-αt(K1cos(sqrt(ω2-α2)t ) + K2sin(sqrt(ω2-α2)t)). Then in order to determine for...

To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation: $$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$ and using the tetrad formalism to compute the coefficients of the...

I'm reading "Differentiable manifolds: A Theoretical Physics Approach" by Castillo and on page 170 of the book a calculation of the Ricci tensor coefficients for a metric is illustrated. In the book the starting point for this method is the equation given by: $$d\theta^i = \Gamma^i_{[jk]}...

Consider the function ##f:[0,1]\rightarrow \mathbb{R}## given by $$f(x)=x^2$$ (1) The Fourier coefficients of ##f## are given by $$\hat{f}(0)=\int^1_0x^2dx=\Big[\frac{x^3}{3}\Big]^1_0=\frac{1}{3}$$ $$\hat{f}(k)=\int^1_0x^2e^{-2\pi i k x}dx$$ Can this second integral be evaluated?

In a self learning project I am fooling around book https://faculty.washington.edu/rjl/fdmbook/ I want to do some of the computation myself to better understand the concepts but the book is Matlab based and Matlab is too expensive. Does anyone by any chance have some of the codes provided by...

Suppose ##a_0+a_1x+\ldots+a_nx^n=0## and restrict the domain of ##p## to the set of real numbers excluding the roots of ##p##. Note that: if ##a_0 == 0##: ##x=0## is a root of ##p## else: ##x=0## is not a root of ##p## Assume the latter. Subtract ##a_0## from both sides of the equation...

https://www.toppr.com/ask/question/if-in-the-expansion-of-1-xn-the-coefficient-of-14th-15th-and-16th/ https://www.sarthaks.com/402983/if-in-the-expansion-of-1-x-n-the-coefficients-of-14-th-15-th-and-16-th-terms-are-in-a-p-find-n Specifically regarding my problem the coefficients of x^8, x^9 and...

Consider polarized light crossing a sharp boundary between two volumes, each of a different but uniform refraction index ##n_1## or ##n_2##. Prove that the sum of the transmission and reflection coefficients of this light ##R+T=1##, where $$R \equiv {I_R \over I_I} = \left( {E_{0_R} \over...

let's take a chemical equation - CH4+2O2 ------> CO2+2H2O From Reactant side- the coefficient of CH4 is 1 and the coefficient of O2 is 2 From Product side - the coefficient of CO2 is 1 and the coefficient of H2O is 2 we can write this chemical equation in terms of molecules,atoms,moles. Can we...

Hi, I am interested in understanding the relationships between Fourier series and Fourier transform better. My goal is 1) Start with a set of ordered numbers representing Fourier coefficients. I chose to create 70 coefficients and set the first 30 to the value 1 and the remaining to zero. 2)...

Assuming the line element ##ds^s=e^{2\alpha}dt^2-e^{2\beta}dr^2-r^2{d\Omega}^2 ##as usual into the form ##ds^s=e^{-2\alpha}dt^2-e^{-2\beta}dr^2-r^2{d\Omega}^2##, I found that the ##G_{tt}## tensor component of first expression do not reconcile with the second one though, it fits for ##G_{rr}...

Let $a,\,b,\,c$ be three distinct integers and $P$ be a polynomial with integer coefficients. Show that in this case the conditions $P(a)=b,\,P(b)=c,\,P(c)=a$ cannot be satisfied simultaneously.

Summary:: Finding the corrected coefficients Suppose you obtained the following magnitude results based off your observations from standard stars: ##\kappa_0 = 0.65##, ##\kappa_1 = 0.10##, ##\alpha_0 = 2.00##, ##\alpha_1 = 0.05##, where ##\kappa_0,\kappa_1## are the extinction coefficients and...

I read about the Steinharts-Hart's equation here and I decided to try and calculate the four coefficients of the extended Steinharts-Hart's equation: $$\frac{1}{T} = A + B \ln(R) + C \cdot (\ln(R))^2 + D\cdot (\ln(R))^3 $$ Where T is a temperature and R is a resistance of the e.g. NTC...

Dear Everyone, I am wondering how to use the integral formula for a holomorphic function at all points except a point that does not exist in function's analyticity. For instance, Let $f$ be defined as $$f(z)=\frac{z}{e^z-i}$$. $f$ is holomorphic everywhere except for $z_n=i\pi/2+2ni\pi$ for...

Given any system with discreet energy eigenstates, φn(x)e-iEnt . The φn are functions only of position. But are they also almost always real-valued?Thanks in advance.

Suppose we know the general form of a multinomial function (for example ##f(x,y,z) = k_1 x^2 y^3 z + k_2 x^3 z^5##) and we have access to a computer program that can evaluate ##f## at arbitrary values (for example, arbitrary values of ##(x,y,z)##) where the coefficients ( for example, ##k_1##...

Hi, I really struggled to dig valuable things out of internet and books related to high order homogeneous differential equation with variable coefficients but I have nothing. All methods I see involves given solution and try to find others(like reduction of order method), even for second order...

Hi, everyone. I'm trying to get the next identity It is in the format <j1, j2; m1, m2 |j, m>. I hope you can help me

Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12$. Find $|f(0)|$.

I am working through David Griffiths' "Introduction to Quantum Mechanics". All of the solutions are provided online by Griffiths himself. This is Problem 2.5(e). I understand his solution but I'm confused about one thing. After normalizing Ψ, we find ##A=\frac {1}{\sqrt2}##. Griffiths notes that...

Hey! :o A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$. I want to calculate $\int_0^{T_0}x^2(t)\, dt$. I have done the following: It holds that...

i tired using complex identity equation for sin(pi*k/3) but it doesn't work out

Einstein coefficients tell us that there is some probability for an atom to go from E_1 to E_2 given by the coefficient of absorption. This is fine, but why is there only one coefficient (absorption) going from E_1 to E_2 and two for the transition E_2 to E_1? Spontaneous emission makes sense...

Hello, I would like to understand know how the lift coefficient ##C_L## versus AoA curve and the drag coefficient ##C_D## versus AoA curve are determined for the various tabulated NACA profiles. Are computer simulation run for the different profiles assuming a certain Reynolds' number? Or can...

%My code: %Type of signal: square T = 40; %Period of the signal [s] F=1/T; % fr D = 23; % length of signal(duration) dt=(D/T)*100; N = 50; %Number of coefficientsw0 = 2*pi/T; %signal pulset1= 0:0.002:T; % original signal sampling x1 = square((2*pi*F)*(t1),dt);%initial square signal t2=...

Hey there! I am current taking an introductory course on PDE's, and our professor hasn't really emphasized last part of solutions from separation of variables. Now its not strictly going to be on the exam, however I remember doing this with ease a few years back, but for some reason now I...