# Cauchy-Riemann equations

Here we follow Spiegel (1964) [1] or Levinson and Redheffer (1970). [2]

Any complex valued function ${\displaystyle f(z)=u(x,y)+iv(x,y)}$ of the complex variable ${\displaystyle z=x+iy}$ that satisfies the following system of first-order partial differential equations, known as the Cauchy-Riemann equations.

${\displaystyle {\frac {\partial u(x,y)}{\partial x}}={\frac {\partial v(x,y)}{\partial y}}\quad \quad {\mbox{and}}\quad \quad {\frac {\partial v(x,y)}{\partial x}}=-{\frac {\partial u(x,y)}{\partial y}}.}$

is called analytic. An analytic function ${\displaystyle f(z)}$ is differentiable with respect to ${\displaystyle z}$.

## Deriving the Cauchy-Riemann equations

Formally we may write the derivative of ${\displaystyle f(z)}$ with respect to ${\displaystyle z}$ using the classical definition of the derivative

${\displaystyle {\frac {df(z)}{dz}}=\lim _{\Delta z\rightarrow 0}{\frac {f(z+\Delta z)-f(z)}{\Delta z}}.}$

Here, of course, we recognize that the theory of limits of complex valued sequences and series can be established.

If we write ${\displaystyle f(z)}$ in terms of its real and imaginary parts, the formal representation of the derivative becomes

${\displaystyle {\frac {df(z)}{dz}}=\lim _{\Delta x\rightarrow 0,\;\Delta y\rightarrow 0}{\frac {u(x+\Delta x,y+\Delta y)-u(x,y)+i(v(x+\Delta x,y+\Delta y)-v(x,y))}{\Delta x+i\Delta y}}}$

where ${\displaystyle \Delta z=\Delta x+i\Delta y.}$

For the derivative to exist, the limit must exist and be unique. Therefore, it should not matter whether we take the ${\displaystyle \lim _{\Delta x\rightarrow 0}}$ first or then ${\displaystyle \lim _{\Delta y\rightarrow 0}}$ second, or vice versa. We must obtain an equivalent result either way.

If we take the ${\displaystyle \lim _{\Delta y\rightarrow 0}}$ first, then the resulting limiting process is

${\displaystyle {\frac {df(z)}{dz}}=\lim _{\Delta x\rightarrow 0}{\frac {u(x+\Delta x,y)-u(x,y)+i(v(x+\Delta x,y)-v(x,y))}{\Delta x}}.}$

Recognizing the partial derivative with respect to ${\displaystyle x}$ we have

${\displaystyle {\frac {df(z)}{dz}}={\frac {\partial u(x,y)}{\partial x}}+i{\frac {\partial v(x,y)}{\partial x}}.}$

Alternatively, we may take the ${\displaystyle \lim _{\Delta x\rightarrow 0}}$ and the resulting limit becomes

${\displaystyle {\frac {df(z)}{dz}}=\lim _{\Delta y\rightarrow 0}{\frac {u(x,y+\Delta y)-u(x,y)+i(v(x,y+\Delta y)-v(x,y))}{i\Delta y}}={\frac {\partial v(x,y)}{\partial y}}-i{\frac {\partial u(x,y)}{\partial y}}.}$

Thus, we have found two ways to represent ${\displaystyle df(z)/dz}$.

Because both of these expressions for ${\displaystyle df/dz}$ must be equivalent, we equate the real and imaginary parts of these expressions to obtain

${\displaystyle {\frac {\partial u(x,y)}{\partial x}}={\frac {\partial v(x,y)}{\partial y}}\quad \quad {\mbox{and}}\quad \quad {\frac {\partial v(x,y)}{\partial x}}=-{\frac {\partial u(x,y)}{\partial y}}.}$

These two equations are known as the Cauchy-Riemann equations. Any complex valued function ${\displaystyle f(z)=u(x,y)+iv(x,y)}$ that satisfies the Cauchy-Riemann equations is called analytic.

Analytic functions are not rare. Most, if not all of the functions encountered applied mathematics likely satisfy satisfy the Cauchy-Riemann equations except possibly at isolated points. These isolated points are poles, branch points, and essential singularities.