Mathquery

Newton's Forward Difference Interpolation Formula:      Let y = f(x) be a function of x and let us suppose that yᵢ = f(xᵢ) ...(1) for i = 1,2,3,.....,n satisfying the condition xᵢ = x₀+ih where  'h' is the interval of difference .        Now our aim is to constuct a function Φ(x) of degree not higher than n such that   Φ(xᵢ) = yᵢ   ............(2)           Since Φ(x) is a polynomial of degree n  then we can write Φ(x) = a₀ + a₁(x-x₀) + a₂(x-x₀)(x-x₁)     +                                 a₃(x-x₀)(x-x₁)(x-x₂)+......                   + aₙ(x-x₀)(x-x₁)(x-x₂).....(x-xₙ₋₁)...(3) Let us find the value of a₀,a₁,a₂.....aₙ satisfying the equations (2) and (3) From equation (2) , we get Φ(x₀) = y₀  From equation (3) , we get Φ(x₀) = a₀ ...

Derivation of Newton's Fundamental Interpolation Formula:     Let y = f (x) be a function with given values yᵢ = f(xᵢ) for (n+1) points x₀,x₁,x₂,.....,xₙ . Our aim is to construct a polynomial Φ(x) of degree not higher than n satisfying the following conditions        Φ(xᵢ) =yᵢ=f(xᵢ) ........(1)                              for i = 0,1,2...,n Let us take the polynomial Φ(x) in the following form  Φ(x) =a₀+a₁(x-x₀)+a₂(x-x₀)(x-x₁)+a₃(x-x₀)(x-x₁)(x-x₂) + ......+aₙ(x-x₀)(x-x₁)(x-x₂)....(x-xₙ)                                                         ........(2) where a₀,a₁,....aₙ i.e aᵢ's are constants to be determined . Putting  i=0 in equation (1) ,we get  Φ(x₀) = y₀ = f(x₀) i.e f(x₀) = Φ(x₀) Again ,...

Lagrangian Interpolation Formula :        Let y = f(x) be a real valued function which is defined in an interval [a,b] . Let  x₀ , x₁ ,x₂ ,............xₙ be n+1 distinct points in that interval at which the respective values y₀ ,y₁,y₂ ............yₙ are tabulated.      Now our aim is to construct a polynomial Φ(x) of degree ≤ n , which interpolates f(x) such that       Φ(xᵢ) = y(xᵢ) , i = 1,2,3,.......,n .........(1)    Let us suppose that the polynomial Φ(x)                                   n be given by Φ(x) = Σ  lᵢ(x) y(xᵢ)  .........(2)                                  i= 0      where each lᵢ(x) is a polynomial of degree ≤n in xᵢ , called  Lagrangian function.      The function ...

Three Theorems of Isomorphism: First Theorem of Isomorphism:              If f: G →G' be an onto homomorphism with kernel K = ker f , then G/K ≈ G'  In other words , every homomorphic image of a group G is isomorphic to a quotient group of G.  Proof :       Define a map Φ : G/K →G' such that ,                                       Φ (Ka) = f(a) , a∈G We Show Φ  is an isomorphism :        That Φ is well defined follows by                            Ka = Kb                       ⇒ab⁻¹ ∈K = Ker f                      ⇒f(ab⁻¹) = e'                ...

Fundamental Theorem Of Homomorphism Of Group :           Let Φ be a homomorphism of G onto G̅ with kernel K .Then G/K ≈ G̅ . Proof:               Let G̅ be the homomorphie image of a group G and Φ be the corresponding homomorphie . Then K is normal subgroup of G .       To prove that G/K ≈ G̅ . If   a ∈ G , the Ka ∈ G /K and Φ(a) ∈G̅  Let ψ : G/K →G̅ such that ψ(Ka) = Φ(a) ∀ a∈G  Where Ka is called right coset and Kb is called left coset. To Show The Mapping ψ is well defined :       i.e if a,b ∈G and Ka = Kb ,then  ψ(Ka) = ψ(Kb)  We have Ka = Kb => ab⁻¹ ∈ K                   => Φ(ab⁻¹) = e̅     (identity of G̅)                =>Φ(a) Φ(b⁻¹) = e̅              => Φ(a)...

Taylor's Theorem and proof: Theorem :                     If f(x,y) is a function which possesses continuous partial derivatives of order n in any domain of a point (a,b) , and the domain is large enough to contain a point (a+h,b+k) with it, then there exists a positive number 0<θ<1, such that  f(a+h,b+k) = f(a,b) + (h∂/∂x + k ∂/∂y)f(a,b)             + (1/2!)(h∂/∂x + k∂/∂y)² f(a,b) + ....+{1/(n-1)!}(h∂/∂x + k∂/∂y)ⁿ⁻¹f(a,b)+Rₙ,  where  Rₙ = (1/n!)(h∂/∂x + k∂/∂y)ⁿf(a+θh,b+θk)                                                        0<θ<1. proof: Let x = a+th , y = b+tk , where 0≤t≤1 is a parameter, and f(x,y) = f(a+th,b+tk) = φ(t) Since the partial derivatives of f(x,y) of order n are continuous in the domain under...

Bessel Equation and Bessel Function: Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number α, the order of the Bessel function.         The differential equation      x²y" + xy' + (x² - p²)y = 0 .......(1)   Where p is a real constant is called BESSEL'S(1784 _ 1846)EQUATION of order p.           It is clear that x=0 is a regular singular point of the equation . Hence we assume a solution of the form          ∞    y = Σ   cₙxⁿ⁺ʳ                           .........(2)         n=0 where c₀≠0 Substitution of the series for y , y'  and y" in equation (1) yields      ∞         ...