Pseudoanalytic function
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In mathematics, pseudoanalytic functions are functions introduced by that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations.
In mathematics, pseudoanalytic functions are functions introduced by that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations. Let z=x+iy and let \sigma(x,y)=\sigma(z) be a real-valued function defined in a bounded domain D . If \sigma>0 and \sigma_x and \sigma_y are Hölder continuous, then \sigma is admissible in D . Further, given a Riemann surface F , if \sigma is admissible for some neighborhood at each point of F , \sigma is admissible on F . The complex-valued function f(z)=u(x,y)+iv(x,y) is pseudoanalytic with respect to an admissible \sigma at the point z_0 if all partial derivatives of u and v exist and satisfy the following conditions: u_x=\sigma(x,y)v_y, \quad u_y=-\sigma(x,y)v_x If f is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.
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