Cauchy-Riemann equations

Here we follow Spiegel (1964) or Levinson and Redheffer (1970).

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

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

is called analytic. An analytic function $f(z)$ is differentiable with respect to $z$.

Deriving the Cauchy-Riemann equations

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

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

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

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

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

where $\Delta z=\Delta x+i\Delta y.$

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

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

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

Recognizing the partial derivative with respect to $x$ we have

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

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

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

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

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

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

These two equations are known as the Cauchy-Riemann equations. Any complex valued function $f(z)=u(x,y)+iv(x,y)$ 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.

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