Mathquery

Green's Theorem:   Statement :-           If a domain E, regular with respect to both the axes , is bounded by a contour C , and f and g are two single - valued functions which along with their partial derivatives ∂f/∂y and ∂g/∂x  are continuous on E , then       ∫∫  (∂g/∂x - ∂f/∂y) dx dy = ∫ (f dx + g dy )        E                                         C  where the line integral is taken in the positive direction . Proof :-         Let us first consider a function f which , alongwith its partial derivative ∂f/∂y ,is continuous on a region E , regular with respect to y-axis . Let E be bounded by contour C , consisting of the curves y= φ(x) , y= ψ(x) , x = a , x = b , such that                  ...

Cauchy - Riemann Equations :- Cauchy-Riemann Equation Necessary Conditions For A Function To Be Analytic : Cauchy-Riemann Equation Statement :- Cauchy-Riemann Equation     The necessary conditions for  w = f(z) = u(x,y) + i v(x,y) to be analytic (differentiable) at any point z = x + i y of its domain D is that the four patial derivatives ∂u/∂x , ∂u/∂y ,∂v/∂x ,∂v/∂y should exists and satisfy the C-R partial differential equations   .          ∂u/∂x = ∂v/∂y  and  ∂u/∂y = - ∂v/∂x Proof :- Cauchy-Riemann Equation       Let f(z) = u(x,y) + i v(x,y) be analytic at any point z of its domain , then               f'(z) = lim     f(z+δz) - f(z) / δz                         δx-->0         exists and is unique . i.e it i...

Lioville's Theorem :- Statement :       If a function f(z) is analytic for all limit values of z and is bounded then f(z) is constant . Proof :        Let z₁ , z₂ be any two point of the z-plane the contour C to be a large circle of radius R centred at origin and containing the point z₁ , z₂ . Therefore |r₁|<R and |z₂|<R . Also as f(z) is bounded therefore ∃ a positive M such that f|(z)|≤M for all z.      By Cauchy's Integral formula we have       f(z₁) = 1/2πi ∫ f(z) dz /z-z₁                           c      f(z₂) = 1/2πi ∫ f(z) dz /z-z₂                           c ∴ f(z₁) - f(z₂) = 1/2πi ∫ f(z) dz /z-z₁                                ...

Morera's Theorem In Complex Analysis :- Morera's Theorem Statement :        If f(z) is continuous in a simple connected domain D and if                     ∫f(z) dz = 0                    c    for every closed path in D, then f(z) is analytic in D .  Morera's Theorem Proof : Morera's Theorem      Let z₀ be a fixed point and z a variable point inside the domain D , then the value of the integral                    z                    ∫ f(z) dz                    z₀  is independent of the curve joining z₀ to z and is a function of the upper limit z . Then we have                   ...

Definition  Of Fourier Transform :         The Fourier Transform of a function f denoted as f̂ is defined by                                  ∞       f̂ (ξ) = 1/√(2π ) ∫ f(x) e ^ -iξx dx    .....(1)                                -∞          whenever the integral on the right exists . It is obvious that the integral on the right of (1) exists if                           ∞                            ∫ |f(x)| dx    exists .                          -∞   If the fourier transform f̂ of a function f is known the functio...

Definition :         A function f is said to be an even function if f(x) = f(-x) for all x and it is odd if   f(-x) = -f(x) . Examples :           Sin kx , x,x³ and any power of x are all odd functions where as Cos kx , x ,1,x² and any even power of x are even functions .     The following properties of even and odd functions are easy to check           a                                      a (i)     ∫ f(x) = 0 if f is odd = 2∫ f(x) dx          -a                                     0                                         ...

Periodic Functions And Fourier Series :- Definition :         A function f is said to be periodic with period T if  (i) f(x) is defined for all x and f(x+T) = f(x) for all  x for some positive number T .      For example , sin x is periodic with period 2π , since sin (x+2π) = sin x . A periodic function has many periods, for if               f(x) = f(x+T) then       f(x) = f(x+T) = f(x+2T) = ....=f(x+nT),    where n is any integer. Hence when T is a period of f, nT is also a period of f , but while referring to period we mean the smallest one .     Since each of the functions sin x , cos x , sin 2x , cos 2x , .... are of period 2π , we may think of representing a given function f of period 2π by an infinite series of these function as                        ∞  ...