For questions on the solving/usage/applications of the Cauchy-Riemann equations. Use this tag alongside the complex analysis tag.
In the field of complex analysis in mathematics, the Cauchy–Riemann equations (or, C-R equations) consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. Cauchy-Riemann Equations are discovered by the French mathematician Augustin Louis Cauchy $(1789-1857)$ and the German mathematician Georg Friedrich Bernhard Riemann $(1826-1866)$.
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- Suppose that $$f(z)=f(x+iy)=u(x,y)+i~v(x,y),$$is differentiable at the point $~z_0=x_0+i~y_0)~$. Then the partial derivatives of $~u,~\text{and}~v~~$ exist at the point $~(x_0,y_0)~$, and can be used to calculate the derivative at $~(x_0,y_0)~$. That is, $$f'(z_0)=u_x(x_0,y_0)+~i~v_x(x_0,y_0)\qquad .. .....(1)$$ and also $$f'(z_0)=u_y(x_0,y_0)+~i~v_y(x_0,y_0)\qquad..........(2)$$ Equating the real and imaginary parts of Equations $(1)$ and $(2)$ gives the so-called Cauchy-Riemann Equations: $$u_x(x_0,y_0)=v_y(x_0,y_0)\qquad \text{and} \qquad u_y(x_0,y_0)=-v_x(x_0,y_0)$$
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C-R equation has a lots of application in Analysis, Fluid Dynamics and many other fields of Mathematics and physics. For more details please find the last two references.
References:
https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations
http://www.ijmetmr.com/oloctober2015/NaisanKhalafMosah-BShankar-A-30.pdf
https://www.quora.com/What-are-the-applications-of-the-Cauchy-Riemann-equations
