What is Coefficients: Definition and 803 Discussions

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression (including variables such as a, b and c). When variables appear in the coefficients, they are often called parameters, and must be clearly distinguished from those representing other variables in an expression.
For example,




2

x

2



x
+
3


{\displaystyle 2x^{2}-x+3}
, has the real coefficients 2, -1, and 3 respectively, and




a

x

2


+
b
x
+
c


{\displaystyle ax^{2}+bx+c}
, has coefficient parameters a, b, and c respectively assuming x is the variable of the equation.
The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the real coefficient 3 and the parameter represented by c.
Similarly, the coefficient attached to the highest multiplicity of the variable in a polynomial is referred to as the leading coefficient. For example in the expressions above, the leading coefficients are 2 and the parameter represented by a.
The binomial coefficients occur in the expanded form of



(
x
+
y

)

n




{\displaystyle (x+y)^{n}}
, and are tabulated in Pascal's triangle.

View More On Wikipedia.org
  1. J

    A Margules' Power Series Formula: Deriving Coefficients

    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...
  2. A

    A The standardized and unstandardized canonical correlation coefficients

    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?
  3. D

    How to Approximate Aerodynamic Coefficients? (Lift/Drag)

    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...
  4. F

    I Linear regression, feature scaling, and regression coefficients

    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...
  5. M

    Engineering How to implement a transfer function in Simulink with variable coefficients?

    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...
  6. F

    A Expression of Shot noise when expanding ##a_{\ell m}## coefficients

    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...
  7. S

    Comp Sci Plot periodic function with Fourier coefficients

    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...
  8. K

    How do I find the coefficients of static friction and kinetic friction?

    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
  9. S

    Characterize Fourier coefficients

    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 ##...
  10. T

    I Second Order ODE with Exponential Coefficients

    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!
  11. link223

    Friction Coefficients: Determining Force Direction?

    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...
  12. L

    B Dividing Lamé coefficients

    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...
  13. A

    I Transformation of connection coefficients

    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
  14. drooble122

    Method of detached coefficients division

    -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...
  15. JD_PM

    Finding specific heat C_p coefficients using NIST

    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##...
  16. Haorong Wu

    I Coefficients in expansions of a vector potential

    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}}...
  17. guyvsdcsniper

    Finding the coefficients of a Resistance vs. Temp curve

    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...
  18. Z

    A Einstein's coefficients recoil effect

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

    How to find the coefficients of this expansion?

    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...
  20. S

    Flat Belt and Pullies with different Coefficients

    I would like to attached a picture here...
  21. C

    A Calculation of Fourier coefficients using SAMBA methodology

    Hello everyone. I have 4 samples of 50 elements from 4 unknown random variables obtained from a Karhunen-Loève decomposition using Matlab's pca (each one is a column of size 50 from the coefficient matrix). I am following the article SAMBA: Sparse Approximation of Moment-Based Arbitrary...
  22. D

    Engineering Question - Calculating Coefficients for 2nd Order Transient Analysis

    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...
  23. snypehype46

    Riemann curvature coefficients using Cartan structure equation

    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...
  24. snypehype46

    I Computing Ricci Tensor Coefficients w/ Tetrad Formalism

    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]}...
  25. docnet

    Find the Fourier Coefficients

    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?
  26. T

    Scientific Computing: Method of Undetermined Coefficients in Python

    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...
  27. Eclair_de_XII

    If a polynomial is identically zero, then all its coefficients are 0

    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...
  28. AN630078

    Coefficients in the expansion of (1+x)n in A.P, find n

    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...
  29. P

    Prove the reflection and transmission coefficients always sum to 1

    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...
  30. S

    Can we write mass in coefficients of elements in a chemical equation

    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...
  31. S

    MATLAB Turning Fourier coefficients into an interpolated freq domain function

    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)...
  32. Bishal Banjara

    I Inverting the metric coefficients in the Schwarzschild line element

    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}...
  33. anemone

    MHB Polynomial with integer coefficients

    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.
  34. L

    Finding the corrected coefficients of your target

    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...
  35. 7

    Steinharts-Hart equation: Calculating the coefficients marathon

    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...
  36. C

    MHB Using Cauchy Integral Formula for Laurent Series Coefficients

    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...
  37. LarryS

    I Are Energy Eigenstate Coefficients Real-valued?

    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.
  38. S

    I Finding multinomial coefficients by evaluation?

    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##...
  39. mertcan

    A Solution to a differential equation with variable coefficients

    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...
  40. victor01

    I Clebsch–Gordan coefficients: An Identity

    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
  41. anemone

    MHB Solving a Third-Degree Polynomial with Real Coefficients

    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 $|f(0)|$.
  42. SamRoss

    What are the coefficients of psi_n for this state?

    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...
  43. M

    MHB Calculate the integral using the Fourier coefficients

    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...
  44. lottotlyl

    Engineering Fourier series coefficients

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

    I How can a photon "stimulate emission" if it really does (Einstein coefficients)

    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...
  46. F

    Lift and drag coefficients vs AoA graphs of a specific NACA profile

    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...
  47. M

    Engineering Reconstruct a signal by determining the N Fourier Coefficients

    %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=...
  48. N

    I Separation of variables - Getting the Fourier coefficients

    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...
Back
Top