7.1 Cauchy's theorem

We begin with a simple observation connecting differentiability of a function of a complex variable to a relation among of partial derivativesof the real and imaginary parts of the function. Actually, we have already visited this point in [link] .

Cauchy-riemann equations

Let $f=u+iv$ be a complex-valued function of a complex variable $z=x+iy\equiv (x,y),$ and suppose $f$ is differentiable, as a function of a complex variable, at the point $c=(a,b).$ Then the following two partial differential equations, known as the Cauchy-Riemann Equations , hold:

and

We know that

and this limit is taken as the complex number $h$ approaches 0. We simply examine this limit for real $h$ 's approaching 0 and then for purely imaginary $h$ 's approaching 0. For real $h$ 's, we have

For purely imaginary $h$ 's, which we write as $h=ik,$ we have

Equating the real and imaginary parts of these two equivalent expressions for $f^{\text{'}}\left(c\right)$ gives the Cauchy-Riemann equations.

As an immediate corollary of this theorem, together with Green's Theorem ( [link] ), we get the following result, which is a special case of what is known as Cauchy's Theorem.

Let $S$ be a piecewise smooth geometric set whose boundary $C_S$ has finite length. Suppose $f$ is a complex-valued function that is continuous on $S$ and differentiable at each point of the interior $S^0$ of $S.$ Then the contour integral ${\int }_{C_S}f\left(\zeta \right)d\zeta =0.$

For future computational purposes, we give the following implications of the Cauchy-Riemann equations.As with [link] , this next theorem mixes the notions of differentiability of a function of a complex variable and the partial derivatives of itsreal and imaginary parts.

Let $f=u+iv$ be a complex-valued function of a complex variable, and suppose that $f$ is differentiable at the point $c=(a,b).$ Let $A$ be the $2\times 2$ matrix

Then:

1. $|f^{\text{'}}{\left(c\right)|}^2=det\left(A\right).$
2. The two vectors
   $${\overrightarrow{V}}_1=(u_x(a,b),u_y(a,b))\text{and}{\overrightarrow{V}}_2=(v_x(a,b),v_y(a,b))$$
   are linearly independent vectors in $R^2$ if and only if $f^{\text{'}}\left(c\right)\ne 0.$
3. The vectors
   $${\overrightarrow{V}}_3=(u_x(a,b),v_x(a,b))\text{and}{\overrightarrow{V}}_4=(u_y(a,b),v_y(a,b))$$
   are linearly independent vectors in $R^2$ if and only if $f^{\text{'}}\left(c\right)\ne 0.$

Using the Cauchy-Riemann equations, we see that the determinant of the matrix $A$ is given by