Riemann Conditions and Analytic Functions

But dz = dx + idy can be obtained in different ways. By taking dz = dx by letting dy = 0) and applying the chain rule of differentiation, then  

For the derivative to exist, these two derivatives must be equal, therefore  

These are the Cauchy-Riemann conditions. Functions that satisfy the Cauchy-Riemann conditions are called analytic functions. The Cauchy-Riemann conditions are satisfied by any analytic function and, hence, any of its successive derivatives. This property of analytic functions makes them useful in the solution of problems in two-dimensional elasticity. 

Considering Eqn. (3.14) and the operation on the real and imaginary parts of an analytic function /(z), one can see that by using the definition of derivatives and the Cauchy-Riemann conditions  

In other words, the real and imaginary parts of analytic functions are harmonic and would satisfy the biharmonic equation (see Eqn. (3.14)). The task then becomes one of identifying the appropriate analytic functions that can satisfy the boundary conditions of the problem. The methodology is applied to the solution of crack problems. 

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