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This Blog is only based on making discussion on different types of mathematical theorems and related solutions.It may useful for those people who want to learn more theorems of different scientists. Here is the collection of all the theorems from primary level, whichay helpful for readers'

         Hello my motive is to clarify and discuss the theorems described by different mathematicians all over the world.

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Cauchy Riemann Equations For Analytic Function

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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...
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PFAFFIAN Differential Equations And It's Solutions

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PFAFFIAN Differential Equations And It's Solutions :-      The relation              n             Σ  Fᵢ(x₁,x₂,......xₙ) dxᵢ = 0            i=1   where Fᵢ's (i=1,2,....n) are functions of all or some of the n independent variables x₁,x₂,.....xₙ is called a Pfaffian differential equation . Here we shall study Pfaffian differential equation in three independent variables which is of the form              P dx + Q dy + R dz = 0    ...........(1)    where P, Q , R are functions of x , y and z . Setting -->                        -->               X = (P,Q,R)  and dr = (dx,dy,dz), equation (1) can be written in the vector notation as            ...
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Fundamental Theorems Of Integral Calculus

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Fundamental Theorems Of Integral Calculus : First Fundamental Theorem Of Integral Calculus : Theorem 1 :                 If a function f is bounded and integrable on [ a,b ] , then the function F defined as                    x                            F(x) = ∫ f(t) dt , a≤x≤b                                         0 is continuous on [ a,b ] and further more , if f is continuous at a point of [ a,b ] , then F is derivable at c and F'(c) = f(c) . Proof :           It is given that the function f is bounded . Then by definition ∃ a number k such that |f(x)|≤ k for x ∈ [ a,b ] ......(1)       Let x₁ , x₂ ∈[ a,b ] such that a≤x₁≤x₂≤b .    ...
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