CauchyRiemaun

Cauchy-Riemann equations are a pair of partial differential equations that relate the real and imaginary parts of a complex function. They are named after the French mathematicians Augustin-Louis Cauchy and Bernhard Riemann. The equations are fundamental in the study of complex analysis and are used to determine whether a function is holomorphic (analytic) in a given region.

For a function f(z) = u(x, y) + iv(x, y), where z = x + iy, the Cauchy-Riemann equations are given by:

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

These equations must be satisfied simultaneously for f(z) to be holomorphic at a point. If the partial derivatives exist and are continuous, and the Cauchy-Riemann equations are satisfied, then f(z) is holomorphic in that region. The equations are also used to find the derivative of a holomorphic function, which is given by:

f'(z) = ∂u/∂x + i∂v/∂x

The Cauchy-Riemann equations have important implications in various areas of mathematics and physics, including fluid dynamics, electromagnetism, and quantum mechanics. They are also used in the study of conformal mappings, which are mappings that preserve angles between curves.