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