The Remarkable World of Special Functions

The world of mathematical special functions is as intriguing and interesting as any domain, say in physics, or the natural world. The trouble is that one usually doesn't appreciate these functions enough while taking Calculus or Linear Algebra or Differential Equations, and one must wait until a research focus brings them to prominence.

Let's take the case of the Bessel functions. In solar physics one very key equation (Helmholtz) for which the (axially symmetric- in cylindrical coordinates r, z, t) Bessel function solution is:

B_z (r) = B_o J_o(a r)

B_t(r)) = B_o J_1(ar)

t = theta

J_0(a r) is a Bessel function of the first kind, order zero and J1 (ar) is a Bessel function of the first kind, order unity. The Bessel functions are defined (cf. Menzel, 'Mathematical Physics', 1961, p. 204):

J_m(x) = (1/ 2^m m!) x^m [1 – x^2/ 2^2 1! (m + 1) + x^4/ 2^4 2! (m +1) (m + 2) -…

-(1)^j x^(2j) / 2^(2j) j! (m + 1) (m +2 )……(m + j) + …]

for m = 0 and m = 1 forms one gets:

J_o(x) = 1 - x^2/ 2^2 (1!)^2 + x^4/ 2^4 (2!)^2 - x^6/ 2^6 (3!)^2 + ......

J_1(x) = x/ 2 - x^3/ 2^3*1! 2! + x^5/ 2^5 *2!3! - x^7/ 2^7 *3!4! - .....


The equations in B_z, B_t, with the special Bessel functions at root, are critical in describing the respective magnetic fields for a magnetic tube.

For a cylindrical magnetic flux tube (such as a sunspot represents viewed in cross-section) the “twist” is defined:

T(r) = (L * B_t(r))/ (r * B_z (r))

Where L denotes the length of the sunspot-flux tube dipole. If the twist value exceeds 2(pi) then the magnetic configuration may be approaching instability and a solar flare.

Then there is the Gamma function (call it 'G') for which:

G(a) = (a - 1 )!

where 'a' is a positive integer.

Thus, for a = 3:

G(3) = (3 - 1)! = 2! = 2*1 = 2

One can also make use of a recursion formula:

G(a + 1) = a G(a)

For example: G(4) = G (3 + 1) = 3 G(3) = 3 (2) = 6

Check this from the earlier formula: G(a) = (a - 1)!

G(4) = (4 - 1)! = 3! = 3*2*1 = 6

Now, there is also the Beta function, call it B(u,v) which can be expressed in terms of the Gamma functions G(u), G(v)

Thus:

B(u,v) = G(u) G(v)/ G(u + v)

Consider the Beta function B(3, 4):

B(3, 4) = G(3) G(4)/ G(3 + 4) = (2) (6)/ G(7) = 12 / G(7)

where: G(7) = (7 - 1)! = 6! = 6*5*4*3*2*1 = 720

so B(3,4) = 12/ 720 = 1/60

Why all the fuss about Beta and Gamma functions here?

Well, if any readers happened to have caught the PBS series 'The Elegant Universe', they'd have seen string theorist Brian Greene scribbling the Euler equation for string theory on a blackboard:

B(p, q) = G (p) G(q)/ G(p + q)

and arriving at the specific string theory form:

Beta([1- alpha(s)][1 - alpha(t)] =

Gamma (1 - alpha(s)) Gamma (1 - alpha(t))/ Gamma(2 - alpha(s) - alpha(t))

Special functions can indeed be loads of fun and have wide applications. In a future foray I'll look at fractional Gamma functions.