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M.KASTHURI-ppt

This power point presentation describes the basic knowledge of analytic functions, harmonic functions and its applications

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M.KASTHURI-ppt

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  1. FUNCTIONS OF COMPLEX VARIABLES Dr. M. KASTHURI, M. Sc., Ph. DAssistant ProfessorDepartment Of MathematicsDKM College For Women(Autonomous)Vellore - 1

  2. Analytic Functions A function f(z) is said to be analytic in a region R of the complex plane if f(z) has a derivative of each point of R and if f(z) is single valued. A function f(z) is said to be analytic at a point z if z is an interior point of some region where f(z) is analytic.

  3. Applications of Analytic Functions Fluid Flow:For a given flow of an incompressible fluid there exist an analytic function f(z) = φ(x,y)+iψ(x,y) where, f(z) is called complex potential of the flow ψ is called stream function and φ is called the velocity potential

  4. C AUCHY – RIEMANN EQUATIONS The Cauchy – Riemann equations on a pair of real valued functions of two real variables u(x,y) and v(x,y) are the two equations:u_x = v_y and u_y = -v_xwhere u and v are the real and imaginary parts of a complex-valued function f(x,y)=u(x,y)+iv(x,y).

  5. Necessary and Sufficient conditions for analyticity A necessary and sufficient conditions for a function f=u+iv to be analytic are that: 1. The four partial derivatives of its real and imaginary parts u_x, u_y, v_x and v_y satisfy the Cauchy-Riemann equations 2. The four partial derivatives mentioned above are continuous

  6. EXAMPLES 1. f(z)=z^3 is analytic2. f(z)=x-iy/x^2+y^2 is analyticabove two functions are satisfied C-R equations and thus they are analytic.Further, f(z)=2xy+i(x^2-y^2) is not satisfying C-R equations and therefore it is not analytic

  7. Harmonic Functions If f(z)=u+iv is an analytic function in some region R, then Cauchy-Riemann equations are satisfied. That implies that both u and v are satisfy Laplace’s equation in two variables, therefore f(z)=u+iv is called as harmonic function. That is,

  8. Harmonic Conjugate If f(z)=u+iv is an analytic function in which u(x,y) is harmonic, then v(x,y) is called as harmonic conjugate of u(x,y). v(x,y) is a harmonic conjugate to u(x,y) if the four partial derivatives u_x, u_y, v_x and v_y are continous and are satisy C-R equations.Examples:1. u(x,y)=x^2-y^2 is a harmonic function2. u(x,y)=x^2-y^2-y is a harmonic function

  9. Applications of Harmonic Functions Electrostatic fields:The force of attraction or repulsion between charged particle is governed by Coloumb’s law. This force can be expressed as the gradient of a function φ, called the electrostatic potential. The electrostatic potential satisfies Laplace’s equation This φ will be the real part of some analytic function F(z)= φ(x,y)+ ψ(x,y) That is,

  10. THANK YOU

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