I have not posted anything in some time due to school stresses. I have had a couple of days off however, but it did not occur to me to post anything until Yesdterday. I almost cried because of what I saw after studying the maclaurin series. Deriving Euler’s formula: e to the pi (i)=-1 was the most awe inspiring thing I have done in my entire life. I do not understand why it works, nor do I think I am capable of understanding it. And I am sure anyone reading this has already derived Euler’s formula. Yet, I must write it here if just for reference to read to myself in the future.
What if we have some function f(x) that is not defined by basic operations (addition or multiplication). We know that, while this function is not defined by basic operations, it must have a polynomial representation. Thus, if we say:
We can see that all we must do is figure out the values for all the c’s.
If we set x=0, then . Then take the derivative of f(x).
And it is easy to see that this could be repeated infinitly and that the denominator a in f^n(x)/a is a factorial.
Thus we have a new representation of f(x):
It’s also interesting to note (just cool, little use I believe):
Some wonderful things about this is it allows us to approximate functions that are not expressed through addition or multiplication or functions involving transcendental numbers with extreme accuracy fairly easly.
Let’s approximate some functions to i=4 at f(0):
Before we continue, notice that cos(x), sin(x) and e^x all look similar. The only thing stoping them from being different is skiping terms and sign changed!
If we add sin and cosine, we remove the skiping:
Now all we need to do to have e^x is change those signs. But how? What function follows that same sign pattern?
Notice now that only the locations corrisponding the sin(x) are multiplied by i.
From this we can conclude (I don’t know how to prove):