Cosmos Computed

I was reading a calculus textbook when I noticed it said that cos( x 2 ) doesn't have an elementary antiderivative. Elementary antiderivative? Clearly, they were hiding something. They didn't say it didn't have an antiderivative; they said it didn't have an elementary  antiderivative. Of course, I wanted to know what the antiderivative was. If it wasn't elementary, it had to be really awesome. I looked up the integral of sin( x 2 ). Turns out, the integral cannot be expressed as anything other than itself. It's known as the Fresnel S integral, is written as S( x ), and is defined as the integral of sin( x 2 ). There's another Fresnel integral known as the Fresnel C integral which is written as C( x ) and defined as the integral of cos( x 2 ). I also saw some graphs of the integrals. One really cool graph involved the parametric equations x  = C( t ) and y  = S( t ), and was called the "Euler spiral." It had a cool spirally shape, and I immediately...

Happy π-day! And happy π-month! Today's month and day - that is, March 14 or 3.14 - includes the first 3 digits of π. And today's month and year - March 2014 or 3.14 - also includes the first 3 digits of π. We won't have another double-day for π for the next 100 years, so enjoy this one! For the special occasion, I'm posting two π-related posts, one for π-month and the other for π-day. In both posts, I'm setting the font size to 16.1527897 pixels, which is approximately π * π + π + π. This is the second post, for π-day; for the first, go to  http://greatmst.blogspot.com/2014/03/pi-month-pi-day-post-1.html . In this post, I will list 5 common myths about π, and explain why they're wrong. 1. Pi is wrong Actually, π is defined as the ratio between the circumference of a circle and its diameter. There really isn't anything about this ratio that could possibly be wrong, because a ratio is a ratio - just like an orange is an orange or a piano is a piano. You can...

The digits of π, organized in a very new way Happy π-day! And happy π-month! Today's month and day - that is, March 14 or 3.14 - includes the first 3 digits of π. And today's month and year - March 2014 or 3.14 - also includes the first 3 digits of π. We won't have another double-day for π for the next 100 years, so enjoy this one! For the special occasion, I'm posting two π-related posts - one for π-month, and the other for π-day. In both posts, I'm setting the font size to approximately π * π + π + π. This is the first post, for π-month; to see the second, go to http://greatmst.blogspot.com/2014/03/pi-month-pi-day-post-2-5-common-pi-myths.html . In this post, I am including an essay I wrote about whether π or τ is the more superior constant. This was written for people who know very little about math, so the basic idea should be easy to understand even for people who are not mathematically inclined. Should Tau Replace Pi? A constant is any number or value that ne...