Mathquery

Periodic Functions Of Fourier Series :              Generally, periodic functions are the functions which returns the same value in regular interval of time . But in trigonometric functions , it returns the same value in the time interval of 2π radian .   For example :               The best example to describe periodic functions is sine function i.e,        sin(x+2π) = sinx         So as discussed before Fourier series is generated by these types of periodic functions like sine and cosine functions . Theorem Related To Periodic Function of Fourier Series :         For a periodic function of period 2π , prove that              β          β+2π     (i)   ∫ f dx = ∫   f dx ,            α  ...

                Fourier Series   Definition :           If the numbers a₀,a₁....aₙ,....,b₁,...bₙ.... are derived from a function f by means of Euler_Fourier formulas :                           π            aₙ = 1/π ∫ f(x) cosnx dx ,n=0,1,2...                         -π                                 .......(1)                          π           bₙ = 1/π ∫ f(x) sinnx dx , n=0,1,2...                         -π then the series                    ...

Taylor's Theorem For Power Series  :   Statement :           Let          ∞                                                               Σ aₙ xⁿ be a power series with                     n= 0  radius of convergence R , and let                       ∞            f(x) = Σ aₙ xⁿ  , |x| < R                      n=0 Then for any a∈ ]-R ,R[ , prove that f can be expanded in a power series about 'a' which converges for  |x-a| < R- |a| , and                ∞     f(x) = Σ...

Weiertrass Approximation Theorem :   Statement :                If f is a real continuous function defined on a closed interval [a,b] then there exists a sequence of real polynomials {Pₙ} which converges uniformly to f(x) on [a,b] i.e lim Pₙ(x)=f(x)                                                    n-->∞ converges uniformly on [a,b] . Proof :        If a=b , the conclusion follows by taking  Pₙ(x) to be a constant polynomial , defined by   Pₙ(x) = f(a) for all n . We may thus assume that a<b . We next observe that a linear transformation                          x' = (x-a) / (b-a) is a continuous mapping of  [a,b] onto [1,0] . Accordingly , we assume without ...

Abel's Theorem For Power Series : Statement :                                  ∞            If the series Σ aₙ is convergent and                                  n=0               ∞ has the sum s , then the series Σ aₙ xⁿ is                                                       n=0 uniformly convergent for 0≤x≤1 and                                   ∞                      lim      Σ aₙ xⁿ = s .         ...

Theorem For Uniform Convergence Of Power Series :           T he power series Σ aₙxⁿ is uniformly convergent for |x| ≤P≤R where R is the radius of convergence . Proof :           Consider a number ρ' between ρ and R . Since , the series is convergent for |x| = ρ' , then by definition there exists k   independent of n so that                 |aₙPⁿ|< k ∀ n       ⇒ for |x| ≤ ρ , |aₙxⁿ| =                             |aₙρⁿ(x/ρ')ⁿ|<(ρ/ρ')ⁿ which is independent of x .        But the series is geometric series with common ratio ρ/ρ' < 1 , therefore the series            kΣ (ρ/ρ')ⁿ is convergent .       Thus , by Weirstrass's M_test the power series is uniformly convergen...