Cauchy-Riemann equations (idea)

The Cauchy-Riemann Equations are the set of relationships between the partial derivatives of a complex-valued function of a complex variable. Whenever they hold at a point, the function is said to be differentiable at that point. If they hold in a disk D around some point (and the partial deriviatives are differentiable within that disk D) the function is said to be analytic at that point. If the Cauchy-Riemann equations are always true (i.e. the function is analytic on C) then the function is said to be an entire function. This is true for most basic elementary functions.

Rectangular Coordinates

Proof of the Cauchy-Riemann equations in rectangular coordinates.

Let f(z) be a complex-valued function of a complex variable

z = x + iy

Let

f(z) = u(z) + iv(z)

Define the derivative of f(z) to be

         lim   f(z + Δz) - f(z)
f'(z) =        ----------------
        Δz->0         Δz

Then we have

         lim   u(z + Δz) - u(z)     v(z + Δz) - v(z)
f'(z) =        ---------------- + i ----------------
        Δz->0         Δz                   Δz

Now we let

Δz = Δx + 0 i

This gives

         lim   u(x + Δx, y) - u(x, y)     v(x + Δx, y) - v(x, y)
f'(z) =        ---------------------- + i ----------------------
        Δx->0            Δx                         Δx

Recalling the definition of a partial derivative from vector calculus shows that

         ∂u     ∂v
f'(z) =  -- + i --
         ∂x     ∂x

Now we return to our previous equation in u, v, and z and let

Δz = 0 + i Δy

This gives

         lim   u(x, y + Δy) - u(x, y)     v(x, y + Δy) - v(x, y)
f'(z) =        ---------------------- + i ----------------------
        Δy->0           i Δy                       i Δy

Again recalling the definition of partial derivative, we see that

            ∂u   ∂v
f'(z) = - i -- + --
            ∂y   ∂y

         OR

         ∂v     ∂u
f'(z) =  -- - i --
         ∂y     ∂y

Observe that these are both equations for f'(z)! Thus we set the real and imaginary parts equal to one another and obtain the famous Cauchy-Riemann equations in rectangular form.

 ∂u   ∂v       ∂u     ∂v
 -- = --  and  -- = - --
 ∂x   ∂y       ∂y     ∂x

Q.E.D.

Polar Coordinates

Proof of the Cauchy-Riemann Equations in polar coordinates.

If we let

z = r (e^iθ);

Then we have the following important relationships which are familiar from analytic geometry

x = r cos θ
y = r sin θ

We proceed to finding polar equivalents of our partial derivatives

∂u   ∂u ∂θ   ∂u     -1
-- = -- -- = -- * -------
∂x   ∂θ ∂x   ∂θ   r sin θ

∂v   ∂v ∂r   ∂v     1
-- = -- -- = -- * -----
∂y   ∂r ∂y   ∂r   sin θ

Since we know these expressions are equal from the rectangular forms

∂u       ∂v
-- = - r --
∂θ       ∂r

Continuing with the next set gives

∂u   ∂u ∂r   ∂u     1
-- = -- -- = -- * -----
∂y   ∂r ∂y   ∂r   sin θ

  ∂v     ∂v ∂θ     ∂v     -1
- -- = - -- -- = - -- * -------
  ∂x     ∂θ ∂x     ∂θ   r sin θ

Again, we know these equations are equal, so

∂v     ∂u
-- = r --
∂θ     ∂r

Thus we have the Cauchy-Riemann Equations for polar coordinates as well!

∂u       ∂v     ∂v     ∂u
-- = - r -- and -- = r --
∂θ       ∂r     ∂θ     ∂r

Q.E.D.

References:

MathWorld.Wolfram.com

George Cain. Complex Analysis. http://www.math.gatech.edu/~cain/winter99/complex.html