We will now explore the notion of differenciation in the complex plane. First a complex-valued function is just like a real-valued function except the input values can be complex numbers and the output values can be complex numbers. Thus for example is a complex-valued function. Note if we write then we can think of as . Similarly any complex-valued function in general can be written as for some real-valued functions . We refer to as the real part of and sometimes write and as the imaginary part of and sometimes write . However, there is a minor problem. Unlike with single or double variable functions which we can graph either by a 2 dimensional or 3 dimensional graph we cannot graph complex-valued functions. Because it has the form so it is a function of two variables and the output is two variables (the real and imaginary part). But that is not a serious problem, we can sometimes graph the real and imaginary parts independently if we really need to.
Let be a complex-valued function defined around some point . We say is differenciable at when exists, or if you perfer . And denote the limit by . Note, here the limit is the limit as we approach the point for all possible paths, unlike in ordinary Calculus which simply means left and right handed this limit has infinitely many different approach and they all need to exist (think of this as a two-variable limit in Calculus 3 if you want to, we are approaching the point from all possible directions). This is why complex-valued that can be differenciated are so much better behaved then in Calculus.
Definition 1: Let be differenciable at a point we say that is analytic at .
I will use the term analytic instead of differenciable to make a distinction between complex analysis derivative and Calculus derivative.
Example 6: Consider let be any point. Now . Thus, just like expected.
It can be show easily that polynomials can be differenciated in the same way as the Power Rule for derivatives. In fact, all the properties of derivatives: sum, difference, product, quotient, and chain rules are all preserved. So basically everything is the same except of we have .
Now we reach a very important theorem whose proof we will omit (again this tutorial is made as simple as possible, writing proofs will make it harder). Before going into the theorem there are something called open sets. An open set is like an open interval something that does not contain its boundary (or its endpoints). For example, is the disk which is open because its boundary is not contained. A closed set is for example which is the disk which does contain its boundary. When we talk about differenciation we talk about it on open sets because it makes no sense to talk about talking the quotient limit on the boundary since the limit needs to exists from all sides. Just like we never talk about a differenciation on a closed interval since it makes no sense to talk about the derivative at because the limit from the left does not exist. Therefore, differenciation of complex-valued functions is always done on open sets. There is just a problem, I did not define mathematically what open sets actually are, I am not going to do that because that requires topology again something which will make this tutorial more difficult. Thus, simply think of open sets as sets in the complex plane which does not contain their boundary. If these open set notion seems confusing I will try to avoid it as much as possible and instead all theorems and computations are going to be done on open disks (for ) centered at of radius .
Theorem 2: Suppose has radius of convergence then on the open disk the function is analytic and its derivative is given by .
We define , , .
We can show have radius of convergence and by Theorem 2 the derivative of is itself, the derivative of is and differenciated is .
The next identity is very important. I am not going to derive it because I probably already seen people do it at least 10 times in person, and it would be really boring for me to do it again. It is one of those things people like to derive. It is easy if you expand everything in infinite series.
Euler Identity: For any complex number we have .
Example 7: We can find that . In fact for any integer . This shows that the equation has infinite many solutions in the complex plane. This also shows there is no such thing as a logarithm because the exponential function is not one-to-one. (There are restricted logarithms functions but we will not worry about them).
There are two more important functions defined by power series: and . These are analytic everywhere too and . They are basically sine and cosine without the alternation of signs. They are called hyperbolic sine and hyperbolic cosine. Note . And it can also be easily shown that and .
We have some properties that should be familar.
Theorem 3: Let be complex numbers and an integer:
Warning! The rule no longer works.
Note and . Thus, and . Or more elegantly in hyperbolic functions and . Make the substitution to get and . Now thus . And thus multiply both sides by to get . This gives us a simple theorem.
Theorem 4: For any complex number we have:
The signifigance of Theorem 4 is for computational work.
Example 8: Let us compute for arbitrary . We can use Theorem 3 property 3 to obtain . Now using Theorem 4 property 3 we get . The reason why this is easier is because now everything is real valued.
Example 9: We will solve the equation . Think of and by previous example it means and . Look at the first equation, since it means and so . But then that means . To solve this equation write multiply both sides by to get . You can use quadradic formula, it cannot be the negative sign otherwise we have a negative on RHS so it must be a plus, thus, . Which means all the solutions are: for .
Thus, now we know how to differenciate polynomials, exponentials, trigonometric, and hyperbolic functions. That is all we will need for further.