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.
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http://www.math.gatech.edu/~cain/winter99/complex.html