Mathquery

        WELCOME TO MATHEMATICS             I n this mathematics session I shall prove that , under suitable conditions, ' the derivative of the integral and the integral of the derivative are equal ' , and consequently , ' the two repeated integrals are equal for continuous functions '.          Leibnitz's Rule In Mathematics:                If f is defined and continuous on the rectangle R = [a,b;c,d] , and if    (i)  fₓ(x,y) exists and is continuous on the rectangle R , and                      d   (ii) g(x) = ∫ f(x,y) dy , for x∈ [a,b]                     c then g is differentiable on  [a,b] and                            d   ...

Fubini's Theorem :-       In the world of mathematics integration plays an important role . So we have discussed the advance form of integration which results the famous theorem Fubini's Theorem. Statement:              If a double integral , I = ∫∫ f dx dy                                                         R   exists over a rectangle R = [a,b;c,d] , and if     d    ∫ f dy also exists , for each fixed x in [a,b] ,    c                                              b      d    then the iterated integral ∫ dx ∫ f dy exists             ...

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...