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A module containing theorems and exercises leading up to 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 + i v be a complex-valued function of a complex variable z = x + i y ( 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:

t i a l u t i a l x ( a , b ) = t i a l v t i a l y ( a , b ) ,

and

t i a l u t i a l y ( a , b ) = - t i a l v t i a l x ( a , b ) .

We know that

f ' ( c ) = lim h 0 f ( c + h ) - f ( c ) h ,

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

f ' ( c ) = f ' ( a + i b ) = lim h 0 f ( a + h + i b ) - f ( a + i b ) h = lim h 0 u ( a + h , b ) + i v ( a + h , b ) - u ( a , b ) - i v ( a , b ) h = lim h 0 u ( a + h , b ) - u ( a , b ) h + i lim h 0 v ( a + h , b ) - v ( a , b ) h = t i a l u t i a l x ( a , b ) + i t i a l v t i a l x ( a , b ) .

For purely imaginary h 's, which we write as h = i k , we have

f ' ( c ) = f ' ( a + i b ) = lim k 0 f ( a + i ( b + k ) ) - f ( a + i b ) i k = lim k 0 u ( a , b + k ) + i v ( a , b + k ) - u ( a , b ) - i v ( a , b ) i k = - i lim k 0 u ( a , b + k ) - u ( a , b ) k + v ( a , b + k ) - v ( a , b ) k = - i t i a l u t i a l y ( a , b ) + t i a l v t i a l y ( a , b ) .

Equating the real and imaginary parts of these two equivalent expressions for f ' ( c ) 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 C S f ( ζ ) d ζ = 0 .

  1. Prove the preceding corollary. See [link] .
  2. Suppose f = u + i v is a differentiable, complex-valued function on an open disk B r ( c ) in C , and assume that the real part u is a constant function. Prove that f is a constant function. Derive the same result assuming that v is a constant function.
  3. Suppose f and g are two differentiable, complex-valued functions on an open disk B r ( c ) in C . Show that, if the real part of f is equal to the real part of g , then there exists a constant k such that f ( z ) = g ( z ) + k , for all z B r ( c ) .

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 + i v 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 × 2 matrix

A = ( u x ( a , b ) v x ( a , b ) u y ( a , b ) v y ( a , b ) ) .

Then:

  1. | f ' ( c ) | 2 = det ( A ) .
  2. The two vectors
    V 1 = ( u x ( a , b ) , u y ( a , b ) ) and V 2 = ( v x ( a , b ) , v y ( a , b ) )
    are linearly independent vectors in R 2 if and only if f ' ( c ) 0 .
  3. The vectors
    V 3 = ( u x ( a , b ) , v x ( a , b ) ) and V 4 = ( u y ( a , b ) , v y ( a , b ) )
    are linearly independent vectors in R 2 if and only if f ' ( c ) 0 .

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

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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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