Cauchy-Riemann equation

Meaning

Noun

• Given a complex-valued function f and real-valued functions u and v such that f(z) = u(z) + iv(z), either of the equations $\backslash textstyle\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; =\; \backslash frac\{\backslash partial\; v\}\{\backslash partial\; y\}$ or $\backslash textstyle\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; =\; -\backslash frac\{\backslash partial\; v\}\{\backslash partial\; x\}$, which together form part of the criteria that f be complex-differentiable.
• The equivalent single equation $\backslash textstyle\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; +\; i\backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\; =\; 0$.

Origin

• Named after mathematicians Augustin Cauchy (1789-1857) and Bernhard Riemann (1826-1866).