Nearly two hundred years after Friedrich Bessel introduced his eponymous functions, expressions for their derivatives with respect to parameters, valid over the double complex plane, have been found.
In this blog we will show and briefly discuss some formerly unknown derivatives of special functions (primarily Bessel and related functions), and explore the history and current status of differentiation by parameters of hypergeometric and other functions. One of the main formulas found (more details below) is a closed form for the first derivative of one of the most popular special functions, the Bessel function J:
April 7, 2016 — Wolfram Blog Team
Authors that choose to incorporate Wolfram technologies into their books are practitioners in a variety of STEM fields. Their work is an invaluable resource of information about the application of Mathematica, the Wolfram Language, and other Wolfram technologies for hobbyists, STEM professionals, and students.
September 10, 2015 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
This statement is in a math textbook, but it is horrifyingly wrong. A statement like “the letters A–Z cannot be matched up with the numbers 1–26″ would be similarly wrong. These two sets of the same size (here, 26) can be matched up as A1, B2, C3, …, Z26. Can the rational numbers be matched up with the integers? Both are infinite, which allows for the tricks of a technique called Hilbert’s hotel, a hotel with infinite numbered rooms that can always make room for one more guest. The Glencoe claim asks if the cardinality of the integers and rationals is the same. Both are , or Aleph-0, which Georg Cantor proved in the 1870s.
July 2, 2015 — Jenna Giuffrida, Content Administrator, Technical Communications and Strategy Group
We’re always on the lookout for new ideas and ways of using the Wolfram Language that our users produce and choose to write about in their books. In this quarter, we have applications that bridge the gap between art and geometry, and demonstrate intuitive data analysis. In addition to writing books, Wolfram welcomes authors to submit articles for publication in The Mathematica Journal, our very own in-house periodical.
June 28, 2015 — Giorgia Fortuna, Consultant, Advanced Research Group
Three months ago the world (or at least the geek world) celebrated Pi Day of the Century (3/14/15…). Today (6/28) is another math day: 2π-day, or Tau Day (2π = 6.28319…).
Some say that Tau Day is really the day to celebrate, and that τ(=2π) should be the most prominent constant, not π. It all started in 2001 with the famous opening line of a watershed essay by Bob Palais, a mathematician at the University of Utah:
“I know it will be called blasphemy by some, but I believe that π is wrong.”
Which has given rise in some circles to the celebration of Tau Day—or, as many people say, the one day on which you are allowed to eat two pies.
But is it true that τ is the better constant? In today’s world, it’s quite easy to test, and the Wolfram Language makes this task much simpler. (Indeed, Michael Trott’s recent blog post on dates in pi—itself inspired by Stephen Wolfram’s Pi Day of the Century post—made much use of the Wolfram Language.) I started by looking at 320,000 preprints from arXiv.org to see in practice how many formulas involve 2π rather than π alone, or other multiples of π.
Here is a WordCloud of some formulas containing 2π:
June 23, 2015 — Michael Trott, Chief Scientist
In a recent blog post, Stephen Wolfram discussed the unique position of this year’s Pi Day of the Century and gave various examples of the occurrences of dates in the (decimal) digits of pi. In this post, I’ll look at the statistics of the distribution of all possible dates/birthdays from the last 100 years within the (first ten million decimal) digits of pi. We will find that 99.998% of all digits occur in a date, and that one finds millions of dates within the first ten million digits of pi.
Here I will concentrate on dates than can be described with a maximum of six digits. This means I’ll be able to uniquely encode all dates between Saturday, March 14, 2015, and Sunday, March 15, 1915—a time range of 36,525 days.
May 29, 2015 — Wolfram Blog Team
This past week, on May 23, 2015, the much loved and respected John F. Nash Jr., along with his wife, Alicia Nash, passed away in a tragic car accident while returning home from his receipt of the 2015 Abel Prize for his work in partial differential equations. The Nobel winner and his wife were the subject of the 2001 Academy Award winning film A Beautiful Mind. Nash’s most famous contribution to mathematics and economics was in the field of game theory, which has enabled others to build on that work and was the focus of the film.
Nash’s long career as a mathematician was marked by both brilliant achievements and terrible struggles with mental illness. Despite his battle with schizophrenia, Nash inspired generations of mathematicians and garnered a stunning array of awards, including the 1994 Nobel Prize in economic sciences, the American Mathematical Society’s 1999 Leroy P. Steele Prize for Seminal Contribution to Research, and the 1978 John von Neumann Theory Prize. We were personally honored in 2003 when Nash presented his work with Mathematica at the International Mathematica Symposium in London.
May 20, 2015 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project
In many areas of mathematics, 1 is the answer. Squaring a number above or below 1 results in a new number that is larger or smaller. Sometimes, determining whether something is “big” is based on whether a largest dimension is greater than 1. For instance, with sides of length 13,800 km, Saturn’s hexagon would be considered big. A “little polygon” is defined as a polygon where 1 is the maximum distance between vertices. In 1975, Ron Graham found the biggest little hexagon, which has more area than the regular hexagon, as shown below. The red diagonals have length 1. All other diagonals (not drawn) are smaller than 1.
April 21, 2015 — Jenna Giuffrida, Content Administrator, Technical Communications and Strategy Group
What do genealogy, linear algebra, and the Raspberry Pi have in common? Not much, but they come together in this diverse and engaging assortment of books by the international community of authors employing Wolfram technologies in their work.
March 12, 2015 — Stephen Wolfram
Pictures from Pi Day now added »
Between Mathematica and Wolfram|Alpha, I’m pretty sure our company has delivered more π to the world than any other organization in history. So of course we have to do something special for Pi Day of the Century.