In the case of real differentiable functions, we have computation rules such as the chain rule, the product rule or even the inverse rule. In the case of complex functions, we have, in fact, precisely the same rules.
Let be complex functions.
If and are complex differentiable in and , the function is complex differentiable in and (linearity of the derivative)
If and are complex differentiable in , then so is (pointwise product with respect to complex multiplication!) and we have the product rule
If is complex differentiable at and is complex differentiable at , then is complex differentiable at and we have the chain rule
If is bijective, complex differentiable in a neighbourhood of and , then is differentiable at (inverse rule)
If are differentiable at and , then (quotient rule)
First note that the maps of addition and multiplication
are continuous; indeed, let for instance be an open ball. Take such that . Now suppose that we have
where is to be determined later. Then we have
where . Upon choosing
we obtain by the triangle inequality
whence is open. If then is an open set, then will also be open, since is the union of open balls and inverse images under a function commute with unions.
The proof for addition is quite similar.
But from these two it follows that if are functions such that
indeed, this follows from the continuity of and at the respective points. In particular, if is constant (say where is a fixed complex number), we get things like
1. Now suppose indeed that ( open, so that we have a neighbourhood around and the derivative is defined in the sense that the direction in which goes to zero doesn't matter) are differentiable at . We will have
4. Let indeed be a bijection between and which is differentiable in a neighbourhood of . By the inverse function theorem, is real-differentiable at , and we have, by the chain rule for real numbers,
( denoting the identity matrix in and the primes (e.g. ) denoting the Jacobian matrices of the functions seen as functions , "" denoting matrix multiplication),
since we may just differentiate the function . However, regarding and as -algebras (or as rings; it doesn't matter for our purposes), we have a morphism of algebras (or rings)
Let be an open subset of the complex plane, and let be a function which is complex differentiable in (that means, in every point of ). Then we call holomorphic in .
If happens to be, in fact, equal to , so that is complex differentiable at every complex number, is called an entire function. We will see examples of entire functions in the chapter on trigonometry, where the exponential, sine and cosine function play central roles. Another important class of entire functions are polynomials.
In algebra, one studies polynomial rings such as , or, more generally, , where is a ring (one then has theorems that "lift" properties of to , eg. if is an integral domain, a UFD or noetherian, then so is ).
Now all elements of are entire functions. This is seen as follows:
Analogous to real analysis (with exactly the same proof), the function is complex differentiable. Thus, any polynomial
( complex coefficients, ie. constants)
is complex differentiable by linearity.
We may also define , an extension of in . This extension turns out to be equal to
From this, there arises a polynomial ring . Let now be any compact subset of the complex plane, or even a bounded subset. Then it is easy to see from direct arguments that with respect to the topology of uniform convergence, is dense in . Alternatively, one finds that