Mathquery

Cauchy's Integral Theorem: Statement :       The theorem is usually formulated for closed paths as follows :        Let U be an open subset of C which is simply connected . Let f: U-->C be a holomorphic  function , and let γ be a rectifiable path in U whose start point is equal to its end point . Then                         ∮ f(z) dz = 0                         γ Proof :       Let us assume that the   partial derivatives  of a holomorphic function are continuous , the Cauchy Integral Theorem can be proved as direct sequence of    Green's Theorem and the fact that the real and imaginary parts of f = u + i v must satisfy the Cauchy - Riemann equations in the region bounded by γ , and moreover in the open neighbourhood U of this region.     ...

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...