NIST Releases Successor to Venerable Handbook of Math Functions

The National Institute of Standards and Technology (NIST) has released the Digital Library of Mathematical Functions (DLMF) and its printed companion, the NIST Handbook of Mathematical Functions, the much-anticipated successors to the agency's most widely cited publication of all time. These reference works contain a comprehensive set of tools useful for specialists who work with mathematical modeling and computation.

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Glimpses Of A New Mathematical World

A new mathematical object was revealed yesterday during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function. These L-functions encode deep underlying connections between many different areas of mathematics.

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DPGraph + Viewer

Although the website is terrible, DPGraph is actually a very neat graphing application.

E-Tutor Graphing Calculator

E-Tutor has a nice, online, graphing calculator for visualizing equations.

WZGrapher Function Grapher

WZGrapher is an easy-to-use and small-footprinted Function Graphing and Calculation Program written in C language, with capabilities to plot both cartesian and polar functions. WZGrapher can also be used to graph numerical solution curves of integrals, to solve numerically and graph ordinary differential equations up to the fifth order, and to calculate value tables (also of ODEs) including the first derivative values.

I Heart Mathematics

Saw this on the web recently and thought it was cool.
Instead of using $\pm\frac{4}{5}$ for the y-intercept though, the golden ratio φ = $\frac{1\pm\sqrt{5}}{2}$ should have been used for a more natural beauty. Also, what would the equation be in 3D?

Gnuplot

Gnuplot is a portable command-line driven interactive data and function plotting utility for UNIX, IBM OS/2, MS Windows, DOS, Macintosh, VMS, Atari and many other platforms. The software is copyrighted but freely distributed (i.e., you don't have to pay for it). It was originally intended as to allow scientists and students to visualize mathematical functions and data. It does this job pretty well, but has grown to support many non-interactive uses, including web scripting and integration as a plotting engine for third-party applications like Octave. Gnuplot has been supported and under development since 1986.

Wolfram Web Resources

In addition to Mathworld, Wolfram Web Resources hosts an extensive list of other sites, including: Demonstrations, Functions, Gallery, Integrals, and Tones,

TiddlyMath

TiddlyMath is a web notebook for mathematics. It allows you to create math formulas, diagrams and graphs to display on the web.

Emprise JavaScript Charts

EJSChart is a 100% pure JavaScript charting solution.

Continuous, Nowhere Differentiable Functions

The Weierstrass function $f_a(x) = sum_(k=1)^infty sin(pi k^a x)/{pi k^a}$ (originally defined for a = 2) is an example of a continuous function, but differentiable only on a set of points of measure zero.

The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that the function f(x) is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates that there is insufficient evidence to decide whether Riemann actually bothered to give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof that every interval of f contains points at which f does not have a finite derivative, and Hardy (1916) proved that it does not have a finite derivative at any irrational and some of the rational points. Gerver (1970) and Smith (1972) subsequently proved that f has a finite derivative (namely, 1/2) at the set of points x = ${2A+1}/{2B+1}$ where A and B are integers. Gerver (1971) then proved that f is not differentiable at any point of the form ${2A}/{2B+1}$ or ${2A+1}/{2B}$. Together with the result of Hardy that f is not differentiable at any irrational value, this completely solved the problem of the differentiability of f.

Since then, a number of mathematicians have developed examples of nowhere continuous, differentiable (CND) functions and it has been shown that these types of functions are the norm (PDF).

One thing that should be obvious is that these functions are not lines in the geometric sense. These are all limits of other continuous functions, that are mostly differentiable, which converge to a nowhere differentiable function.

In Euclidean geometry, a polygon (or straight line) should only be drawn by pencil and straighthedge. If we were to draw any line, then the straighthedge can be omitted and a person only needs to trace the pencil from the starting point to the finish using whatever path they like. Assuming that the person isn't drunk, the line should be of finite length since we are drawing a line fron start to finish. Thus, any line segment that is a subset of that line, must have an even smaller finite length.

However, with CND functions, if you are to pick a point with a pencil and try to trace it to any other point, you will not go anywhere. The reason being is that the standard definition of a line is not met with the CND function. Since at any point, the function is not differentiable, the line would not know which direction to take. Thus, it would appear to vibrate like an electron held motionless in place.

Therein lies the beauty of such functions. They exist and are continuous, but are not real in the geometric sense.