Let's do two functions today. As sometimes happens, in this case we're not so interested in the functions themselves as the fact that these functions happen to be part of a general class of functions. Just as we can classify the real numbers as even, odd, or neither (numbers like pi, 1/2, and the rest of the non-integers are neither), we can classify functions as odd, even, or neither. This is a random even function and its graph:
What makes this function even? A function is even if it's symmetric about the y-axis. In other words an even function will have the same value at at x = 1 as it does at x = -1, and correspondingly for all other values of x. To say the same thing symbolically, a function is even if and only if
Not so bad. Odd functions are similar, and here's an example:
Odd functions are antisymmetric about the y axis. Along the lines of even functions, symbolically you can say that a function is odd if and only if:
Odd and even function tend to follow similar though not identical patterns to odd and even numbers. An even function times an even function is an even function. Odd times odd is even. Odd times even is odd. For instance, if you take our even and our odd functions above and multiply them, you'll get an odd function with a graph like this:
All this is vaguely interesting, but really why bother with it? Mathematically there's a number of reasons. Understanding the properties of even and odd functions can help simplify a lot of problems. For instance, that last odd function integrated from minus infinity to infinity is zero because the area below the curve on the right is exactly balanced by the area above the curve on the left. We know that without actually having to do the integral by hand. There's a number of other important properties along those lines that can make your life easier.
But from a physics standpoint these symmetry and antisymmetry properties are even more important. The symmetric or antisymmetric character of a wavefunction under exchange of particles is the fundamental difference between bosons and fermions, leading to such important phenomena as the Pauli exclusion principle. We're going to go into some detail about that soon, which is why we're laying the groundwork now. Until then, enjoy the weekend!
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ooo! good topic! can't wait for the next post.
Is gravitation even-function? Newton's r^2 and Green's paired squares, Einstein's ten equations. Obviously even.
A parity Eotvos experiment tests for odd-function gravitation. Oppose chemically identical opposite parity atomic mass distributions as enantiomorphic space groups P3(1)21 versus P3(2)21 or P3(1) versus P3(2). The former contains quartz, berlinite and analogues, cinnabar, tellurium, selenium, benzil. Quartz is densely packed, 12.557 A^3/atom. In P3(1) / P3(2), the gamma-polymorph of glycine is more densely packed, 7.869 A^3/atom.
Quantized gravitations require supplementing Einstein-Hilbert action with a parity-violating Chern-Simons term providing mass to the gauge field. Einstein alone won't quantize - why would that be? Somebody should look.
Great post! What software/program are you using to draw those graphs? It looks pretty neat!
I'm using the new Mathematica. The older versions graphed a not-very-pretty B/W image, but the newer one does great visual work and is very customizable.