The Cauchy-Riemann equations can be derived purely analytically via a simple calculation as above, but they are fundamentally geometric, albeit not in a way that's immediately obvious. The complex numbers can be defined algebraically; the imaginary unit is a root of the polynomial x2 + 1, but the real appreciation of the complex numbers is essentially geometric. In fact, that's exactly how the complex numbers first lost their mysterious aura that made down-to-earth mathematicians so uneasy and earned i the "imaginary" moniker. When Karl Friedrich Gauss introduced the complex plane and showed that the complex numbers can be understood in terms of the geometry of this plane, complex numbers entered mainstream mathematics and nobody had any reservations about what they really meant anymore. In an attempt to relive great moments in mathematical history, I shall now attempt to do something similar and to explain the Cauchy-Riemann equations as geometrically as I can, and in order to do that, I'm going to first need to explain a little more about the geometry of the underlying complex numbers. Linear algebra is a wonderful modern language for talking geometry, and the language I shall adopt below.

Complex multiplication as a linear operation

Let us identify, as Gauss did, the complex plane C with the real plane R2. A complex z number is uniquely determined by an ordered pair (x, y), i.e. z = x + yi, and vice-versa. Under this point of view, complex numbers are simply two-dimensional vectors of real numbers, and addition and subtraction of complex numbers can be accomplished simply by adding and subtracting corresponding entries of these vectors. You can also scale complex numbers by real numbers, and thus we have a bona fide vector space. So far, so good.

The real geometric magic of the complex numbers comes into play when we consider complex multiplication. Let z and w be arbitrary complex numbers, seen as elements of a vector space, r is some arbitrary real number, and let c be a fixed complex number. We shall view c as a linear operator. Really, all of these numbers are complex numbers, they are exactly the same sort of creatures, but I am giving them different roles right now in order to discover an underlying geometric truth. Because the complex numbers are a field, in particular they satisfy the associative and distributive properties that any field has to satisfy. This means that

c(z + w) = c(z) + c(w)
c(rz) = rc(z),

that is, c is indeed a linear transformation; it acts linearly on the vector space of complex numbers. This trivial observation has a nice consequence once we express c as a matrix in the standard basis of {(1,0), (0,1)} of R2, which corresponds to the real and imaginary units of C. Recall that the matrix of a linear transformation in a specific basis is found by seeing what the linear transformation does to such a basis. If c = a + bi and we keep in mind both views of complex numbers as a field and complex numbers as a vector space, then

              /[ 1 ]\                           [ a ]
     c(1) = c( [   ] ) = (a + bi)(1) = a + bi = [   ]
              \[ 0 ]/                           [ b ]
              /[ 0 ]\                            [-b ]
     c(i) = c( [   ] ) = (a + bi)(i) = -b + ai = [   ],
              \[ 1 ]/                            [ a ]

so that in the standard basis of R2, when c = a + bi is viewed as a linear transformation, its matrix in this basis is

          [ a  -b ]
     c =  [       ].
          [ b   a ]

We can take this one step further. Matrices can also be added and subtracted, not just multiplied, and because the determinant of this matrix is a2 + b2 which is zero only if both a and b are zero, all these matrices except the zero matrix are also invertible, which is to say that we can divide. This means that we can view complex numbers as matrices!

Theorem The complex numbers are isomorphic to the field of matrices of the form
          [ a  -b ]
          [       ].
          [ b   a ]

Proof: This is a routine calculation. We would have to prove that matrix addition and subtraction corresponds to complex addition and subtraction, but this is obvious, because these matrices are determined only by two entries which are added and subtracted componentwise just as complex numbers are added and subtracted. It is a little less obvious that matrix multiplication corresponds to complex multiplication, although the view of these matrices as linear maps helps. We can either multiply two arbitrary matrices of this form and see that the result does indeed coincide with the result of multiplying the two corresponding complex numbers, or we can view complex multiplication as a linear transformation, and recall that matrix multiplication corresponds to composition of linear transformations which in turn corresponds to complex multiplication. This second viewpoint also allows us to see that matrix inversion corresponds to complex division, or alternatively we could carry out the computations of matrix inversion and see that they correspond with the computations for complex division. These are all simple verifications, and I don't have any qualms in trusting my readers to carry them out for themselves if they really want to believe this isomorphism. ♦

We could say more here. We could analyse more about the geometry of the complex numbers by analysing the geometry of these matrices. For example, the norm of a complex number when seen as a vector, is square root of the determinants of these matrices, and if you'll notice the similar shape of these matrices with rotation matrices

          [ cos θ   -sin θ ]
          [                ],
          [ sin θ    cos θ ]

then we could also talk about the relationship between complex multiplication and rotations of the complex plane. I shall not pursue these ideas, however, because they are not essential in order to understand the geometry of the Cauchy-Riemann equations. Complex numbers are matrices, and this will do for now.

Complex differentiation as a Jacobian

Let us now talk about functions. Since the complex numbers are a normed field (the norm of a complex number is simply the distance to the origin when viewing complex numbers as vectors), it makes sense to define complex differentiation as follows.

Definition Let f : CC be a complex-valued function. We say that f is differentiable at c if the limit
             f(c+h) - f(c)
       lim  --------------
       h→0        h
exists. In such a situation, we let f'(c) denote such a limit and call it the derivative of f at c.

The definition is exactly the same as the one for real functions. That's what makes it so magical. It seems to be asking a function to be exactly the same as what a corresponding real-valued function would be, but it turns out that the geometry of the complex numbers yields a great bounty out of this definition that the real numbers cannot give. Part of this bounty is the Cauchy-Riemann equations.

Let us go back to the identification of the complex plane C with the real two-dimensional vector space R2. In such a situation, then we are talking about functions f : R2R2, and it also makes sense to talk about differentiation for those functions. For a function of two variables that takes values in two variables, the corresponding idea to the derivative is the Jacobian matrix. Then if we write a complex-valued function f(z) as a function of two variables with two component functions u and v, that is, f(x,y) = (u(x,y ), v(x,y)), the derivative of f is the Jacobian matrix

          [ux uy] 
          [     ],
          [vx vy]   

where I have used subindices in order to represent partial differentiation with respect to a particular variable.

This is the crux of the matter. It turns out that if we view the complex plane as R2, then this Jacobian derivative has to be the same as the complex derivative f'(z). They are both linear transformations (see above why complex numbers can be seen as linear transformations), and they are both the best linear approximation to f, and there is only one derivative, only one best linear approximation. It means that these two objects must be the same. The Jacobian matrix has to be a complex number:

          [ux uy]       [ a  -b ]
   f'(z)= [     ]   =   [       ].
          [vx vy]       [ b   a ]

In other symbols, using now a different notation to denote partial differentiation,

      --  =  --        -- = - -- ,

which are exactly the Cauchy-Riemann equations.

It's remarkable how it all fits in together. Complex multiplication gives us a way to view to view complex numbers on the real plane as linear transformations. Linear transformations have a matrix, and the structure of complex multiplication endows this matrix with a special structure of its own. A complex derivative is also a linear transformation, therefore must also have the same matrix structure. But saying that the complex derivative has the same matrix structure as any other complex number is exactly what the Cauchy-Riemann equations are saying.