the indicial equation
/ in-DISH-ul /
When you launch a Frobenius series y = x^r (a_0 + a_1 x + ...) at a regular singular point, the first thing the equation hands back is not the coefficients but the allowed values of the leading exponent r. The indicial equation is that gatekeeper: a small algebraic equation — usually a simple quadratic in r — whose roots are the exponents r that a valid Frobenius solution may start with. It plays the same role for singular points that the characteristic equation plays for constant-coefficient equations: a little algebra that unlocks the solution's form.
How it arises, step by step. You substitute the Frobenius series into y'' + P(x) y' + Q(x) y = 0 (in the form with the leading factor). Because a_0 is nonzero, the very lowest power of x — namely x^(r - 2) or whatever the lowest is — must have a zero coefficient on its own, and that coefficient is a_0 times a polynomial in r. Setting that polynomial to zero gives the indicial equation. In the clean standard form, if you write x^2 y'' + x p(x) y' + q(x) y = 0 with p_0 = p(0) and q_0 = q(0), the indicial equation is r(r - 1) + p_0 r + q_0 = 0, a quadratic with two roots r_1 and r_2.
Those two roots are the whole plot of what follows. They tell you the leading behaviour of each solution near the singular point — does it stay finite, blow up, or pick up a fractional power — and, crucially, the difference r_1 - r_2 decides which of the three Frobenius cases you are in, including whether a logarithm must appear in the second solution. Read the indicial roots first; they forecast everything.
Bessel's equation x^2 y'' + x y' + (x^2 - nu^2) y = 0 has p_0 = 1, q_0 = -nu^2 (from the constant terms), so the indicial equation is r(r-1) + r - nu^2 = r^2 - nu^2 = 0, giving roots r = +nu and r = -nu. The two roots +-nu are exactly the orders of the two Bessel solutions.
Only the constant terms p_0, q_0 of the (suitably scaled) coefficients enter the indicial equation — the rest of the series does not affect the exponents.
The indicial equation depends only on the values of the coefficients at the singular point, not on their full Taylor expansions — so it is quick to read off, but it tells you only the leading exponents, never the whole solution.