Cauchy-Riemann equations

In complex analysis, the Cauchy-Riemann equations are two partial differential equations that a complex function must satisfy in some region of the complex plane for it to be analytic in that region. This has important implications for the differentiability of complex functions. If we express a complex number z in terms of two real numbers, x and y, through z=x+iy, then a complex function, f(z), can be expressed in terms of two functions of x and y, u(x,y) and v(x,y), as,

$f(z)=u(x,y)+iv(x,y)$

The Cauchy-Riemann equations can then be expressed in terms of u(x,y) and v(x,y) as,

${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},\qquad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$

They are named after Augustin-Louis Cauchy and Bernhard Riemann.

Derivation

When defining the derivative of a real function, it is done through a limit. The same method can be used for complex differentiation, defining it as,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z} }

However we require the derivative and so this limit to be unique. But in the complex plane we can approach the limit in many different ways (e.g. along the real axis or the imaginary axis among others). These must all give the same result. If Δz=Δx+iΔy and expressing f in terms of u(x,y) and v(x,y) as before then the limit becomes,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \lim_{\Delta x, \Delta y \to 0} \left[ \frac{u(x+\Delta x, y+\Delta y) + v(x+\Delta x, y+\Delta y) - u(x,y) - v(x,y)}{\Delta x + i\Delta y} \right] }

Consider the cases of Δz being purely real or purely imaginary, corresponding to taking the limit along wither the real or imaginary axes. If Δz is real then Δy is zero meaning,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \lim_{\Delta x \to 0} \left[\frac{u(x+\Delta x,y) - u(x,y)}{\Delta x} + i \frac{v(x+\Delta x,y) - v(x,y)}{\Delta x} \right] =\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} }

If instead the case of Δz being imaginary (Δx is zero) is considered then,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \lim_{\Delta y \to 0} \left[\frac{u(x,y+\Delta y) - u(x,y)}{i\Delta y} + i \frac{v(x,y+\Delta y) - v(x,y)}{i\Delta y} \right] =\frac{1}{i} \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y} }

For f to be differentiable at a point z these two must be identical. Comparing real and imaginary parts gives the Cauchy-Riemann equations,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} }

See also

• Complex analysis
• Complex number