Playing around with Taylor series and complex numbers…

How complex numbers and calculus can be used to come up with a rather unusual definition for the world famous constant Pi.

This summer was my last before matriculating at university where I plan to pursue an Honors degree in Mathematics. I did not want to work much during this summer; I wanted to savor it because I knew I would possibly never get so much free time again. As many of you can imagine, it eventually got boring.

When it reached that point, I decided to start doing Math for fun — I was hoping to enter a good head space before university. I ended up learning so much and I started venturing on my own. Last week, I was reading some of the proofs on some series for Pi. This got me wondering if I, with all that I had learnt, could come up with my own series for Pi. So my exploration started… I shall take you all through my mashup of complex numbers and taylor series to come up with a definition for Pi that I have not seen before (maybe it already exists).

I started by considering the different functions that yield Pi at some point. Of course I steered away from the trigonometric functions because I was looking for something new — something different. I thought of the natural logarithm. I knew that in the realm of complex numbers, the natural log function could yield values with Pi. First, I needed a convenient value so that I could form the series. I considered the following: