Pseudoanalytic function
In mathematics, pseudoanalytic functions are functions introduced by  that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations. Definitions 
Let  and let  be a real-valued function defined in a bounded domain . If  and  and  are Hölder continuous, then  is admissible in . Further, given a Riemann surface , if  is admissible for some neighborhood at each point of ,  is admissible on . The complex-valued function  is pseudoanalytic with respect to an admissible  at the point  if all partial derivatives of  and  exist and satisfy the following conditions:

If  is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain. Similarities to analytic functions 
 If  is not the constant , then the zeroes of  are all isolated. Therefore, any analytic continuation of  is unique. Examples 
 Complex constants are pseudoanalytic. Any linear combination with real coefficients of pseudoanalytic functions is pseudoanalytic. See also 

 Quasiconformal mapping
 Elliptic partial differential equations
 Cauchy-Riemann equations

References

Further reading 

 
 
 

Complex analysis
Partial differential equations
Types of functions