Once upon a time I was confused about what really made a holomorphic function different from a function on which is differentiable in the real sense. I mean obviously I knew how they were different but I wanted to put it in a context that was more meaningful for me, and after some thinking I thought the following and was satisfied:
Let’s cover some definitions; it’s likely I’ll write these classic definitions in more detail in later posts. Let be a complex function and let be a complex number. is called (complex) differentiable at if the limit
exists, where the limit is taken along whatever path you like. The function is differentiable on a subset if it is differentiable for all . Now a complex number can be written as , where , and a complex function can be written as where and are both functions from to . As -vector spaces, and are isomorphic and we can think of as a function given by
Now we it can be said that a complex function is complex differentiable if and only if it is differentiable in the real sense (i.e. all partial derivatives of and exist and are continuous) and and satisfy the Cauchy-Riemann equations ( and ).
This observation is well known and does a good job of contrasting the two senses of differentiability, but I wanted to know the answer to “why is the derivative of the real version of the function a linear transformation (given by the Jacobian matrix) but the derivative of the complex version another complex function?” The answer to that is obvious in hindsight, but I was happy when I went from not knowing to knowing so I thought I better recount it here. The trick is, of course, that we construct a field isomorphism (which I’ll call ) from into (the ring of 2 by 2 matrices with real entries) given by
Now taking the derivative of gives us a function , but the Cauchy-Riemann equations mean that if is complex differentiable, then . So in that case we can think of as a function from to , and so my question is resolved! The real version of the function is complex differentiable precisely when the Jacobian matrix can be identified with a complex function through the isomorphism .